Question: Let $r\in\{2,3,4\}$, let $\mathcal W_{\underline a}$ be the fully reduced web basis of $\mathsf{Inv}(V_q^{\underline a})$, and let $\underline C=(\underline c_1,\ldots,\underline c_m)$ be a clasp sequence on the boundary vertices of a web, so the boundary vertices are partitioned into interval blocks called clasps, with clasp weights $\lambda_i=\mathrm{wt}(\underline c_i)$. The associated clasping map is $\pi_{\underline C}:\mathsf{Inv}(V_q^{\underline a})\to \mathsf{Inv}(\bigotimes_{i=1}^m V_q(\lambda_i))$. For $[W]_q\in \mathcal W_{\underline a}$, suppose $[W]_q\notin \ker \pi_{\underline C}$ and the corresponding fully reduced hourglass plabic graph $W$ is non-convex with respect to $\underline C$, meaning that for every $i\in[m]$ and every $\underline c_i$-cut path $\gamma$, one has $\mathrm{wt}(\gamma)\ge \mathrm{wt}(\underline c_i)$ in the dominance order on dominant weights. (Here, trips are the distinguished walks in the hourglass plabic graph.) Which statement is equivalent to both of these conditions, and what basis statement also holds for the associated clasped web invariants?

$W$ has no trips that start and end in the same clasp, and the set $\{\pi_{\underline C}([W]_q): [W]_q\in \mathcal W_{\underline a}\ \text{and }W\text{ has no trips that start and end in the same clasp}\}$ forms a basis of $\mathsf{Inv}(\bigotimes_{i=1}^m V_q(\lambda_i))$.

Choices:
B. $W$ has no trips that start and end in the same clasp, and the set $\{\pi_{\underline C}([W]_q): [W]_q\in \mathcal W_{\underline a}\ \text{and }W\text{ is non-convex for every possible clasp sequence on }\underline a\}$ forms a basis of $\mathsf{Inv}(\bigotimes_{i=1}^m V_q(\lambda_i))$.
C. $W$ has no trips that start and end in the same clasp, and the set $\{\pi_{\underline C}([W]_q): [W]_q\in \mathcal W_{\underline a}\ \text{and }W\text{ has no trips that start and end in the same clasp}\}$ spans $\mathsf{Inv}(\bigotimes_{i=1}^m V_q(\lambda_i))$.
D. $W$ has no trips that start and end in the same clasp, and the set $\{\pi_{\underline C}([W]_q): [W]_q\in \mathcal W_{\underline a}\ \text{and }W\text{ has no trips that start and end in the same boundary interval}\}$ forms a basis of $\mathsf{Inv}(\bigotimes_{i=1}^m V_q(\lambda_i))$.
E. $W$ has no trips whose endpoints lie in consecutive clasps, and the set $\{\pi_{\underline C}([W]_q): [W]_q\in \mathcal W_{\underline a}\ \text{and }W\text{ has no trips whose endpoints lie in consecutive clasps}\}$ forms a basis of $\mathsf{Inv}(\bigotimes_{i=1}^m V_q(\lambda_i))$.

same clasp replaced by same boundary interval


[\autoref{thm_main}]
	\label{thmA}
	Let $X$ be an equidimensional projective scheme over a field $\mathbb{k}$.
	Let $\mathscr{L}$ be an ample line bundle on $X$ and $W \subseteq Z$ be two closed subschemes of $X$.
	Then the following three conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $\II_W$ is integral over $\II_Z$.
		\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.
		\item $\deg_\mathscr{L}\left(s^i(Z, X)\right) = \deg_\mathscr{L}\left(s^i(W, X)\right)$ for all $i \ge 0$.
	\end{enumerate}

\begin{headcor}[\autoref{thm_main_zeta}]
	\label{corB}
	Let $I \subseteq J \subset R$ be two homogeneous ideals in a polynomial ring $R = \kk[x_0, \ldots,x_n]$.
	Then the following two conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $J$ is integral over $I$.
		\item $\zeta_I(t) = \zeta_J(t)$.
	\end{enumerate}
\end{headcor}

\begin{headthm}[\autoref{thm_main}]
	\label{thmA}
	Let $X$ be an equidimensional projective scheme over a field $\kk$.
	Let $\LL$ be an ample line bundle on $X$ and $W \subseteq Z$ be two closed subschemes of $X$.
	Then the following three conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $\II_W$ is integral over $\II_Z$.
		\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.
		\item $\deg_\LL\left(s^i(Z, X)\right) = \deg_\LL\left(s^i(W, X)\right)$ for all $i \ge 0$.
	\end{enumerate}
\end{headthm}

\begin{corollary}
	\label{thm_main_zeta}
	Let $J \subset R = \kk[x_0,\ldots,x_n]$ be a homogeneous ideal such that $J \supseteq I$.
	Then the following two conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $J$ is integral over $I$.
		\item $\zeta_I(t) = \zeta_J(t)$.
	\end{enumerate}
\end{corollary}

The \emph{Segre class} associated with an embedding of schemes plays a central role in Fulton--MacPherson's intersection theory and underlies numerous applications.
A detailed account of Segre classes and their applications can be found in \cite{FULTON_INTER,ALUFFI_SURVEY}.

A central and defining property of Segre classes is their \emph{birational invariance}.
Indeed, if $Z$ is a proper closed subvariety of a variety $X$ over a field $\mathbb{k}$, this invariance yields
$$
s(Z, X) \;=\; \eta_*\left(s(E, \mathcal{B})\right) \;=\; \eta_*\left(c\left(N_E\mathcal{B}\right)^{-1} \smallfrown [E]\right) \;=\; \eta_*\left(\frac{[E]}{1+E}\right) \;\in\; A_*(Z),
$$
where $\pi : \mathcal{B} = \Bl_Z(X) \rightarrow X$ is the blow-up of $X$ along $Z$, $\eta : E = E_ZX \rightarrow Z$ is the exceptional divisor, and $N_E\mathcal{B}$ is the normal bundle to $E$ in $\mathcal{B}$ (see \cite[\S 4.2]{FULTON_INTER}).
Thus, by birational invariance, computing Segre classes reduces to the case of effective Cartier divisors, whose Segre classes admit an explicit formula since they are regularly embedded.

The birational invariance of Segre classes also implies that they depend only on the \emph{integral closure} of the underlying ideal sheaf.
More precisely, if we let $Z'= V\left(\overline{\II_Z}\right) \subset X$ where $\overline{\II_Z} \subset \OO_X$ is the integral closure of the ideal sheaf $\II_Z$ of $Z$, then the Segre class $s(Z, X) \in A_*(Z)$ of $Z$ equals the Segre class $s(Z', X) \in A_*(Z')$ of $Z'$ (since $Z'_{\rm red} = Z_{\rm red}$, we can identify their Chow groups).

\smallskip

Our main result demonstrates that Segre classes, in fact, determine integral dependence, leading to the following criterion:

\begin{theorem}
	\label{thm_main}
	Assume \autoref{setup_segre_int_dep}.
	Then the following five conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $\II_W$ is integral over $\II_Z$.
		\item ${j_Z}_*\left(s(Z, X)\right) = {j_W}_*\left(s(W, X)\right)$ in $A_*(X)$.
		\item $\deg_\LL\left(s^i(Z, X)\right) = \deg_\LL\left(s^i(W, X)\right)$ for all $i \ge 0$.
		\item $\deg_\LL\left(\nu^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\nu^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
		\item $\deg_\LL\left(\beta^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\beta^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
	\end{enumerate}
\end{theorem}

\begin{corollary}
 \label{cor_numerical_snapper}
 The following statements hold:
 \begin{enumerate}[\rm (i)]
 \item The terms of $P_{\LL, Z}(m, n)$ of degree $d$ are given as follows
 $$
 P_{\LL, Z}(m, n) \;=\; \sum_{i=0}^d \frac{\deg_\LL\left(\beta^i(\underline{\sigma}, X)\right)}{i!(d-i)!} \,m^{d-i} n^{i} \;+\; \text{{\rm(}lower degree terms{\rm)}}.
 $$
 \item For all $m \in \ZZ$ and $n \gg 0$, we have the equality
 $$
 P_{\LL, Z}(m, n) \;=\; \chi\big(\II_Z^n \otimes \LL^{\otimes (m+n)}\big).
 $$
 \item If $\LL$ is very ample on $X$, we get the equality
 $$
 P_{\LL, Z}(m, n) \;=\; h^0\big(X, \II_Z^n \otimes \LL^{\otimes (m+n)}\big)
 $$
 for all $m \gg 0$ and $n \gg 0$.
 \end{enumerate}
\end{corollary}
\begin{proof}
 (i) By \cite[Theorem B.7, Definition B.8]{KLEIMAN_PICARD}, we have
 $$
 \chi\left(\pi^*(\LL)^{\otimes m} \otimes \OO_{\mathcal{B}^\LL}(n)\right) \;=\; \sum_{i=0}^d \frac{a_{i}}{i!(d-i)!} \,m^{d-i} n^{i} \;+\; \text{(lower degree terms)}
 $$
 where $a_{i}= \int c_1\left(\pi^*(\LL)\right)^{d-i} c_1\left(\OO_{\mathcal{B}^\LL}(1)\right)^i \smallfrown \left[\mathcal{B}\right]$.
 Then the equalities
 $$
 a_i \;=\; \int c_1\left(\LL\right)^{d-i} \smallfrown \pi_*\left( c_1\left(\OO_{\mathcal{B}^\LL}(1)\right)^{i} \smallfrown \left[\mathcal{B}\right]\right) \;=\; \deg_\LL\left(\beta^i(\underline{\sigma}, X)\right)
 $$
 follow from the projection formula and \autoref{prop_blow_up_formula}.

\begin{lemma}
 \label{lem_num_equiv}
 The following conditions are equivalent:
 \begin{enumerate}[\;\;\rm (a)]
 \item $\deg_\LL\left(s^i(Z, X)\right) = \deg_\LL\left(s^i(W, X)\right)$ for all $i \ge 0$.
 \item $\deg_\LL\left(\nu^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\nu^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
 \item $\deg_\LL\left(\beta^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\beta^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
 \end{enumerate}
\end{lemma}
\begin{proof}
 (a) $\Rightarrow$ (b):
 By utilizing \autoref{eq_vogel_Z}, we obtain
 \begin{align*}
 \deg_\LL\left(\nu^i(\underline{\sigma}, X)\right) &\;=\; \int c_1(\LL)^{d-i} \smallfrown \nu^i(\underline{\sigma}, X) \\
 &\;=\; \sum_{j=0}^i\binom{i-1}{j-1} p^{i-j} \int c_1(\LL)^{d-i} c_1(\LL)^{i-j} \smallfrown s^j(Z, X) \\
 &\;=\; \sum_{j=0}^i\binom{i-1}{j-1} p^{i-j} \deg_\LL\left(s^j(Z, X)\right).
 \end{align*}
 Similarly, \autoref{eq_vogel_W} yields
 $$
 \deg_\LL\left(\nu^i(\underline{\eta}, X)\right) \;=\; \sum_{j=0}^i\binom{i-1}{j-1} p^{i-j} \deg_\LL\left(s^j(W, X)\right).
 $$
 So the implication (a) $\Rightarrow$ (b) follows.

\begin{theorem}
 \label{thm_main}
 Assume \autoref{setup_segre_int_dep}.
 Then the following five conditions are equivalent:
 \begin{enumerate}[\;\;\rm (a)]
 \item $\II_W$ is integral over $\II_Z$.
 \item ${j_Z}_*\left(s(Z, X)\right) = {j_W}_*\left(s(W, X)\right)$ in $A_*(X)$.
 \item $\deg_\LL\left(s^i(Z, X)\right) = \deg_\LL\left(s^i(W, X)\right)$ for all $i \ge 0$.
 \item $\deg_\LL\left(\nu^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\nu^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
 \item $\deg_\LL\left(\beta^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\beta^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
 \end{enumerate}
\end{theorem}
\begin{proof}
 (a) $\Rightarrow$ (b): This implication is a consequence of the birational invariance of Segre classes.
 For completeness, we include a short argument.
 Assume that $\overline{\II_Z} = \overline{\II_W}$.
 Thus $Z$ and $W$ have the same support, and so we may identify their Chow groups.
 Let $X_1,\ldots,X_k$ be the irreducible components of $X$ and $m_i$ be the geometric multiplicity of $X_i$ in $X$.
 Let $Z_i = Z \cap X_i$ and $W_i = W \cap X_i$.
 By \cite[Lemma 4.2]{FULTON_INTER}, we have $s(Z, X) = \sum_{i=1}^k m_i s(Z_i, X_i) \in A_*(Z)$ and $s(W, X) = \sum_{i=1}^k m_i s(W_i, X_i) \in A_*(W)$.
 Thus to show the equality $s(Z, X) = s(W, X) \in A_*(Z) = A_*(W)$, we may assume that $X$ is irreducible.

\begin{proposition}
 \label{prop_hard}
 Assume that $\Rees(\II_W)$ is not of finite type as a module over $\Rees(\II_Z)$.
 Then we have the following strict inequality
 $$
 \lim_{n \to \infty} \frac{h^0\left(X, \II_W^{n}/\II_Z^{n} \otimes \LL^{\otimes 2pn} \right)}{n^d} \;>\; 0.
 $$
\end{proposition}
\begin{proof}
 To simplify notation, let
 $$
 \sC \;:=\; \Rees^{\LL^{\otimes p}}(\II_Z) \;=\; \bigoplus_{n\ge 0} \II_Z^n \otimes \LL^{\otimes pn} \quad \text{ and } \quad \sD \;:=\; \Rees^{\LL^{\otimes p}}(\II_W) \;=\; \bigoplus_{n\ge 0} \II_W^n \otimes \LL^{\otimes pn}.
 $$
 Let $\pi : \sY = \fProj_X\left(\sD\right) \rightarrow X$ be the natural projection.
 We cannot have that $W_{\rm red} = X_{\rm red}$, because otherwise the ideal sheaves $\II_Z \subseteq \II_W$ would be nilpotent, and so they would have the same integral closure (equal to the nilradical ideal sheaf of $X$).
 It then follows that $\sY$ is equidimensional of dimension $d$ (see \autoref{rem_blowup_irr_comp}).
 By assumption, we have that $\fProj_X\left(\sD/\sC_+\sD\right) \neq \varnothing$, where $\sC_+\sD \subset \sD$ denotes the graded ideal in $\sD$ generated by $\sC_+ = \bigoplus_{n\ge 1} \sC_n$.
 Let $y \in \sY$ be a closed point lying in $\fProj_X\left(\sD/\sC_+\sD\right)$.
 Let $\II_y \subset \OO_\sY$ be the ideal sheaf of $y$. Let $\aaa \subset \sD$ be a graded ideal in $\sD$ whose sheafification is $\II_y = \widetilde{\aaa}$ and such that $\aaa \supset \sC_+\sD$.
 Since $\pi$ is a closed morphism, the closed point $y \in \sY$ is mapped to a closed point $x = \pi(y)$ in $X$.

\begin{example}[Line bundles that are big and nef do not suffice in \autoref{thm_main}]
 \label{examp}
 Assume $\kk$ is algebraically closed.
 Consider the following two successive blow-ups:
 \begin{enumerate}[\rm (i)]
 \item Let $\pi_1 : Y = \Bl_{p}\left(\PP_\kk^3\right) \rightarrow \PP_\kk^3$ be the blow-up of $\PP_{\kk}^3$ along a closed point $p \in \PP_\kk^3$.
 Notice that the exceptional divisor $E_p\PP_\kk^3$ of $Y=\Bl_{p}\left(\PP_\kk^3\right)$ is isomorphic to $\PP_{\kk}^2$.
 \item Choose a line $C$ in $\PP_\kk^2 \cong E_p\PP_\kk^3$.
 Let $\pi_2 : X = \Bl_C(Y) \rightarrow Y$ be the blow-up of $Y$ along $C$.
 Notice that $X$ is a smooth threefold.
 Let $E = E_CY$ be the exceptional divisor of $X = \Bl_C(Y)$.
 \end{enumerate}
 Let $\pi = \pi_1 \circ \pi_2 : X \rightarrow \PP_\kk^3$ be the natural projection.
 Let $\mathcal{L} = \pi^*\big(\OO_{\PP_\kk^3}(1)\big)$ be the line bundle on $X$ given as the pullback of $\OO_{\PP_\kk^3}(1)$.
 Notice that $\mathcal{L}$ is big and nef.
 Let $k \ge 2$ and consider the effective Cartier divisor $kE$.
 As closed subschemes, we have $E \subsetneq kE \subset X$ (although they have the same support).
 We also obtain that their Segre classes are different
 $$
 s(E, X) \;=\; E - E^2 + E^3 \;\neq\; kE -k^2E^2 + k^3E^3 \;=\; s(kE, X);
 $$
 here $E^i$ denotes the self-intersection of $E$
 (see \cite[Proposition 4.1(a)]{FULTON_INTER}).
 However, we can compute the vanishings
 $$
 \deg_{\mathcal{L}}\left(E\right) \;=\; \deg_{\mathcal{L}}\left(E^2\right) \;=\; \deg_{\mathcal{L}}\left(E^3\right) \;=\; 0.
 $$
 This shows that the big and nef line bundle $\mathcal{L}$ cannot be used in \autoref{thm_main}.
\end{example}
\begin{proof}
 We only need to show the claimed vanishings.
 The projection formula yields
 $$
 \deg_\mathcal{L}(E) \;=\; \int c_1\big(\OO_{\PP_\kk^3}(1)\big)^2 \smallfrown \pi_*\left([E]\right) \quad \text{ and } \quad \deg_\mathcal{L}(E^2) \;=\; \int c_1\big(\OO_{\PP_\kk^3}(1)\big) \smallfrown \pi_*\left(E \cdot [E]\right).
 $$
 By construction, $E$ is mapped onto the point $p$ via $\pi$.
 Since $[E] \in A_2(X)$ and $E\cdot [E] \in A_1(X)$, by dimension reasons and the definition of proper pushforward, we get $\pi_*([E]) = 0$ and $\pi_*(E \cdot [E]) = 0$ (see \cite[\S 1.4]{FULTON_INTER}).
 It then follows that $\deg_{\mathcal{L}}\left(E\right) = \deg_{\mathcal{L}}\left(E^2\right) = 0$.

In \autoref{thmA}, even if one is not interested in integral dependence, the equivalence (b) $\Leftrightarrow$ (c) seems interesting in its own right.
We note that \autoref{thmA} is sharp in the sense that the ampleness hypothesis appears to be indispensable.
\autoref{examp} illustrates this by exhibiting a big and nef line bundle for which the conclusion of \autoref{thmA} does not hold.

The proof of \autoref{thmA} relies on van Gastel’s foundational result \cite{VanGastel}, which establishes the connection between Fulton–MacPherson’s intersection theory \cite{FULTON_INTER} and St\"uckrad–Vogel’s intersection theory \cite{VOGEL,FOV}.
Indeed, our starting point is van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes.
An advantage of considering the Vogel cycle is that it opens the door to applying technical positivity results, including the one established in \autoref{prop_hard}.

\smallskip

As an application of \autoref{thmA}, we show that Aluffi’s Segre zeta function \cite{aluffi2017segre} furnishes a criterion for the integral dependence of homogeneous ideals in a polynomial ring.
This power series records information about the Segre classes that arise when the ideal is extended to projective spaces of arbitrarily large dimension (see \autoref{sect_zeta} for further details).
We obtain the following corollary:

\begin{headcor}[\autoref{thm_main_zeta}]
	\label{corB}
	Let $I \subseteq J \subset R$ be two homogeneous ideals in a polynomial ring $R = \mathbb{k}[x_0, \ldots,x_n]$.
	Then the following two conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $J$ is integral over $I$.
		\item $\zeta_I(t) = \zeta_J(t)$.
	\end{enumerate}
\end{headcor}

The idea of detecting integral dependence through numerical invariants originates in Rees’ seminal work \cite{REES}.
Rees proved that in an equidimensional and universally catenary Noetherian local ring $(R, \mm)$, two $\mm$-primary ideals $I \subseteq J$ have the same integral closure if and only if they share the same Hilbert–Samuel multiplicity.
Since then, the search for numerical criteria to characterize integral dependence has been an important research topic in algebraic geometry, commutative algebra and singularity
theory.
Much effort has gone into extending Rees’ theorem to arbitrary ideals, modules, and more broadly, to algebras (see \cite{BOEGER,TEISSIER_CYC,TEISSIER_RES2, KLEIMAN_THORUP_GEOM,KLEIMAN_THORUP_MIXED, GG, FLENNER_MANARESI,SUV_MULT, GAFFNEY, CIU, UV_CRIT_MOD,     UV_NUM_CRIT,   PTUV, cid2023relative, cidruiz2024polar, CRPU2, BLQ, das2024numerical}).
In this direction, two notable results are:
\begin{itemize}[\;--]
	\item Teissier's Principle of Specialization of Integral Dependence (PSID) \cite{TEISSIER_CYC}, \cite[Appendice I]{TEISSIER_RES2}.
	Given a map of germs of analytic spaces,  the PSID asserts that
	for a family of zero-dimensional ideals with constant Hilbert-Samuel multiplicity, a section is integrally
	dependent on the total family if and only if it is integrally dependent along the fibers corresponding to a dense open  subset of the base.
	\item An extension of Rees’ theorem to arbitrary ideals (in an equidimensional and universally catenary Noetherian local ring $(R, \mm)$) was established by Polini, Trung, Ulrich, and Validashti \cite{PTUV}, using the multiplicity sequence introduced by Achilles and Manaresi \cite{ACH_MANA}. (In the analytic setting, an analogous result had been obtained earlier by Gaffney and Gassler \cite{GG}.)
\end{itemize}
Note that these criteria are local in nature, whereas \autoref{thmA} furnishes a global criterion.

\medskip

  \noindent
\textbf{Outline.}
The structure of the paper is as follows.
In \autoref{sect_prelim}, we introduce the notation used throughout and recall the necessary preliminary results.
\autoref{sect_van_Gastel} develops the relation between Segre classes and Vogel cycles and shows how this connection can be applied.
The proof of \autoref{thmA} is given in \autoref{sect_main}.
Finally, \autoref{sect_zeta} contains the proof of \autoref{corB}.

The post-theorem introduction indicates that the proof of \autoref{thmA} \emph{relies on van Gastel’s foundational result} \cite{VanGastel}, which "establishes the connection between Fulton–MacPherson’s intersection theory" \cite{FULTON_INTER} and "St\"uckrad–Vogel’s intersection theory" \cite{VOGEL,FOV}. The stated \emph{starting point} is "van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes." The introduction further explains the strategy/advantage: "considering the Vogel cycle" allows one to apply "technical positivity results, including the one established in \autoref{prop_hard}." (The full proof is deferred: "The proof of \autoref{thmA} is given in \autoref{sect_main}.")

The post-theorem introduction indicates that, in establishing the main theorem, the proof \emph{relies on van Gastel’s foundational result} \cite{VanGastel}, which "establishes the connection between Fulton–MacPherson’s intersection theory" \cite{FULTON_INTER} and "St\"uckrad–Vogel’s intersection theory" \cite{VOGEL,FOV}. The stated \emph{starting point} is "van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes." The introduction further explains the strategy/advantage: "considering the Vogel cycle" allows one to apply "technical positivity results, including the one established in the following proposition." \begin{proposition}
	\label{prop_hard}
	Assume that $\Rees(\II_W)$ is not of finite type as a module over $\Rees(\II_Z)$.
	 Then we have the following strict inequality
	 $$
	 \lim_{n \to \infty} \frac{h^0\left(X, \II_W^{n}/\II_Z^{n} \otimes \LL^{\otimes 2pn} \right)}{n^d} \;>\; 0.
	 $$
\end{proposition} (The full proof is deferred: the proof of the main theorem is given later.)

To prove the main theorem, let $X$ be an equidimensional projective scheme over a field $\mathbb{k}$.
	\label{thmA}
	Let $\mathscr{L}$ be an ample line bundle on $X$ and $W \subseteq Z$ be two closed subschemes of $X$.
	Then the following three conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $\II_W$ is integral over $\II_Z$.
		\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.
		\item $\deg_\mathscr{L}\left(s^i(Z, X)\right) = \deg_\mathscr{L}\left(s^i(W, X)\right)$ for all $i \ge 0$.
	\end{enumerate}

Biconditional or Equivalence
Classification or Bijection

Question: Let $X$ be an equidimensional projective scheme over a field $\mathbb{k}$, let $\mathscr{L}$ be an ample line bundle on $X$, and let $W \subseteq Z$ be closed subschemes of $X$. Suppose that the ideal sheaf $\mathcal{I}_W$ is integral over $\mathcal{I}_Z$, and that the Segre classes of $Z$ and $W$ in $X$ agree after pushforward to the Chow group $A_*(X)$; that is, $s(Z,X)=s(W,X)$ in $A_*(X)$. If $s^i(Z,X)$ and $s^i(W,X)$ denote the codimension-$i$ components of these Segre classes, which of the following is the precise numerical condition appearing in the theorem and equivalent to those assumptions?

For every $i\ge 0$, the $\mathscr{L}$-degrees of the codimension-$i$ Segre components agree: $$\deg_{\mathscr{L}}\big(s^i(Z,X)\big)=\deg_{\mathscr{L}}\big(s^i(W,X)\big).$$

Choices:
B. There exists an integer $N\ge 0$ such that for every $i\ge N$, the $\mathscr{L}$-degrees of the codimension-$i$ Segre components agree: $$\deg_{\mathscr{L}}\big(s^i(Z,X)\big)=\deg_{\mathscr{L}}\big(s^i(W,X)\big).$$
C. For every $i\ge 0$, if $$\deg_{\mathscr{L}}\big(s^i(Z,X)\big)\ne \deg_{\mathscr{L}}\big(s^i(W,X)\big),$$ then neither $\mathcal{I}_W$ is integral over $\mathcal{I}_Z$ nor $s(Z,X)=s(W,X)$ in $A_*(X)$.
D. For every ample line bundle $\mathscr{M}$ on $X$ and every $i\ge 0$, the $\mathscr{M}$-degrees of the codimension-$i$ Segre components agree: $$\deg_{\mathscr{M}}\big(s^i(Z,X)\big)=\deg_{\mathscr{M}}\big(s^i(W,X)\big).$$
E. For every $i\ge 0$, the $\mathscr{L}$-degrees of the codimension-$i$ Segre components agree, and moreover the ideal sheaves have the same integral closure in $\mathcal{O}_X$: $$\deg_{\mathscr{L}}\big(s^i(Z,X)\big)=\deg_{\mathscr{L}}\big(s^i(W,X)\big),\qquad \overline{\mathcal{I}_W}=\overline{\mathcal{I}_Z}.$$


\begin{theorem}\label{thm:main-dem}
	Let $\eps \in (0,2^{-10})$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
	Let $G$ be an $n$-vertex $(\eps,p)$-typical graph and $k = \eps 2^{-10}p^{-1} \log pn$.
	Then, the following holds for all sufficiently large $n$. With probability at least $1- n^{-2^{-11}\log pn}$, in the random greedy independent set process, we have
	\[ d_i(v) = (1 \pm f_i)\td_i\]
	for all $v \in V_i$ and all $i \in [k-1]$.
\end{theorem}

\begin{theorem}\label{thm:main-prague}
	Let\footnote{In all our statements, $\eps$ is a fixed constant and $n$ tends to infinity.} $\eps \in (0,1)$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
	Then, with high probability we have
	\begin{align*}
		\mathrm{pdim}(G_{n,p}) = O_{\eps} \left ( \dfrac{pn \log n}{\log pn} \right ).
	\end{align*}
\end{theorem}

Motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem~\cite{Deuber,ErdosHajnalPosa,Rodl},
Nešetřil, Pultr and Rödl~\cite{Nesetril1977-pj,Nesetril1978-km} introduced the notion of the Prague dimension of a graph in the 1970s.
For a graph $G$, the Prague dimension $\text{pdim}(G)$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph.
This is closely related to $\text{idim}(G)$, defined as the minimum number of partial clique factors whose union equals $\overline{G}$, the complement of $G$.
In fact,
Nešetřil and Rödl~\cite{Nesetril1978-km} showed that for every graph $G$, we have
\begin{align}\label{eq:pdim-idim}
 \mathrm{idim}(G) \le \mathrm{pdim}(G) \le \mathrm{idim}(G) + 1.
\end{align}

In 1977, Lovász, Nešetřil, and Pultr~\cite{Lovasz1980-kw} showed that a graph on $n\ge2$ vertices has Prague dimension at most $n-1$ and classified all graphs which achieve this bound.
General upper and lower bounds were obtained by Alon~\cite{Alon1986-oc}, who proved that if $G$ is a graph on $n$ vertices with minimum degree at least 1 and maximum degree $\Delta$, then $\log (n/ \Delta)/ \log 2 \le \text{pdim}(G) = O(\Delta^2\log n)$; all our asymptotic expressions are meant as $n \to \infty$, and all logarithms are natural.
The upper bound was later improved by Eaton and Rödl~\cite{Eaton1996-ap}, who showed that
\begin{align*}
 \text{pdim}(G) = O(\Delta \log n).
\end{align*}
Moreover, they proved that this is best possible up to a factor of $1/(\log \Delta + \log \log (n/2\Delta) )$.
When $\Delta$ is a constant, these results provide the order of magnitude of $\text{pdim}(G)$.
Precise asymptotics are known for matchings, trees and cycles~\cite{Alon1986-oc,Lovasz1980-kw,Poljak1981-lt}, hypercubes~\cite{Krivka} and Kneser graphs $K(n,k)$ for constant values of $k$~\cite{Poljak1978-um}.
For more details, we refer the reader to~\cite{Kantor}.

Nešetřil and Rödl~\cite{Nesetril1983-ju} pioneered the study of the Prague dimension of random graphs by showing that the Prague dimension of $G_{n,p}$ is $\Omega_p(n/\log n)$ with high probability for constant values of $p$.
Many years later, Guo, Patton and Warnke~\cite{Guo2023-wq} managed to obtain a matching upper bound by using a Pippenger--Spencer~\cite{Pippenger1989-ak} type
edge-colouring result for random hypergraphs with large uniformities.
More recently, Molnar, Rödl, Sales and Schacht~\cite{MolnarRodlSalesSchacht} studied the Prague dimension of sparse random graphs.
Via a double counting argument, they showed that for $p \gg n^{-2}$, we have\footnote{For any two functions $f$ and $g$, we write $f \ll g$ to mean $f = o(g)$.}
\begin{align}\label{eq:lower-bound-MRSS}
 \text{pdim}(G_{n,p}) \ge \dfrac{pn \log \frac{1}{p} }{5\log n }
\end{align}
with high probability.
Furthermore, by analysing the distribution of independent sets in~$G_{n,p}$, they showed that for all $\varepsilon > 0$ and $p$ such that $n^{-1/3} \log^{4/3} n \ll p \ll n^{-\varepsilon}$, we have $\text{pdim}(G_{n,p}) = \Theta_{\varepsilon}(pn)$ with high probability.
In our first main result, we prove an upper bound that is valid for almost the entire range of $p$.

Nešetřil and Rödl~\cite{Nesetril1983-ju} pioneered the study of the Prague dimension of random graphs by showing that the Prague dimension of $G_{n,p}$ is $\Omega_p(n/\log n)$ with high probability for constant values of $p$.
Many years later, Guo, Patton and Warnke~\cite{Guo2023-wq} managed to obtain a matching upper bound by using a Pippenger--Spencer~\cite{Pippenger1989-ak} type
edge-colouring result for random hypergraphs with large uniformities.
More recently, Molnar, Rödl, Sales and Schacht~\cite{MolnarRodlSalesSchacht} studied the Prague dimension of sparse random graphs.
Via a double counting argument, they showed that for $p \gg n^{-2}$, we have\footnote{For any two functions $f$ and $g$, we write $f \ll g$ to mean $f = o(g)$.}
\begin{align}\label{eq:lower-bound-MRSS}
 \text{pdim}(G_{n,p}) \ge \dfrac{pn \log \frac{1}{p} }{5\log n }
\end{align}
with high probability.
Furthermore, by analysing the distribution of independent sets in~$G_{n,p}$, they showed that for all $\eps > 0$ and $p$ such that $n^{-1/3} \log^{4/3} n \ll p \ll n^{-\eps}$, we have $\text{pdim}(G_{n,p}) = \Theta_{\eps}(pn)$ with high probability.
In our first main result, we prove an upper bound that is valid for almost the entire range of $p$.

In particular, by combining our result with the lower bound in~\eqref{eq:lower-bound-MRSS}, we obtain that for all $\eps > 0$ and $p$ such that $ n^{-1+\eps} \le p \le n^{-\eps}$, with high probability the Prague dimension of $G_{n,p}$ is $\Theta_{\eps}(pn)$.
As it is crucial for our analysis to have concentration on the degrees of all vertices in $G_{n,p}$, we note that $n^{-1}\log n$ is a natural barrier for the range of $p$ our method can handle.

Our methods also allow us to bound the minimum number of independent sets needed to cover all the non-edges of $G=G_{n,p}$, denoted by $\overline{\theta}_1(G)$.
Initially, one may conjecture that with high probability $\overline{\theta}_1(G)$ should be of order $n^2/\alpha(G)^2$ for all values of $p$, which corresponds to the size of a `near' optimal clique cover of $\overline{G}$.
Frieze and Reed~\cite{Frieze1995-xa}, by improving the results of Bollobás, Erd\H{o}s, Spencer, and West~\cite{Bollobas1993-nl}, showed that this is indeed the case for constant values of $p$.
Surprisingly, this is not true for the sparse range.
In~\cite{Guo2023-wq}, Guo, Patton and Warnke proved that for any fixed $\eps >0$ and $n^{-1} \ll p \le 1-\eps$, with high probability we have
\begin{align*}
 \overline{\theta}_1(G_{n,p}) = \Omega \left ( \dfrac{n^2 \log \frac{1}{p} }{\alpha^2(G_{n,p})} \right ).
\end{align*}
It is worth mentioning that due to the work of Matula~\cite{Matula1970-ka}, Bollobás and Erd\H{o}s~\cite{Bollobas1976-al}, Grimmett and McDiarmid~\cite{Grimmett1975-vs}, and Frieze~\cite{Frieze1990-cu}, we know that the size of the largest independent set in $G_{n,p}$ is of order $\Theta(p^{-1}\log pn)$ for $n^{-1} \ll p \le 1- \eps$, for any fixed $\eps > 0$.
Also note that
$\log \frac{1}{p} = \Theta_{\eps}(\log n)$ for $ n^{-1}\ll p \ll n^{-\eps}$.

\begin{theorem}\label{thm:main}
 Let $\eps\in (0,1)$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
 Then, with high probability we have
 \begin{align*}
 \overline{\theta}_1(G_{n,p}) = \Theta_{\eps} \left ( \dfrac{n^2\log n}{\alpha^2(G_{n,p})} \right ).
 \end{align*}
\end{theorem}

\begin{theorem}\label{thm:prob-non-edge}
 Let $\eps \in (0,2^{-10})$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
 Let $G$ be an $n$-vertex $(\eps,p)$-typical graph and $k = \eps 2^{-10} p^{-1} \log pn $.
 Then, for all distinct $u, v \in V(G)$ such that $uv \notin E(G)$, we have
 \begin{align*}
 \Pr{v \in I_k} = \big(1+o_{\eps}(1) \big ) \dfrac{k}{n} \qquad
 \text{and} \qquad \Pr{u,v \in I_k} = \big (1+o_{\eps}(1) \big ) \left ( \dfrac{k}{n} \right )^2.
 \end{align*}
\end{theorem}

Let $\big(J_{i,j}: i \in [t], j \in [s] \big )$ be a collection of $ts$ independent copies of $I_k = I_k(G)$.
 For each $i \in [t]$, define $\mathcal{I}_i$ to be the collection of vertex-disjoint subsets of $V(G)$ given by
 \begin{align*}
 I_{i,j} \coloneqq J_{i,j} \setminus \big ( J_{i,1} \cup \cdots \cup J_{i,j-1} \big )
 \end{align*}
 for all $j \in [s]$.
 By~\eqref{eq:pdim-idim}, it suffices to show that with high probability every non-edge of $G$ is covered by some $I_{i,j}$.
 For $i \in [t]$, the probability that a given non-edge $uv$ is covered by an independent set in the partition $\mathcal{I}_i$ is equal to
 \begin{align}\label{eq:prob-non-edge}
 \sum_{j=1}^{s}\Pr{u,v \in I_{i,j}} & = \sum_{j=1}^{s} \Pr{u,v \in J_{i,j}} \prod_{\ell\le j-1} \Pr{ \{ u,v \} \cap J_{i,\ell} = \emptyset}\nonumber\\
 & = \sum_{j=1}^{s} \Pr{u,v \in I_k} \Pr{ \{ u,v \} \cap I_k = \emptyset}^{j-1}.
 \end{align}
 By the inclusion-exclusion principle, the probability that neither $u$ nor $v$ is covered by $I_k$ is
 \begin{align}\label{eq:prob-non-edge-2}
 \Pr{ \{ u,v \} \cap I_k = \emptyset} &= 1 - \Pr{u \in I_k} - \Pr{v \in I_k} + \Pr{u,v \in I_k} \nonumber \\
 & = 1 - \Theta_{\eps} \left ( \dfrac{k}{n} \right ) = 1 - \Theta_{\eps}(s^{-1}).
 \end{align}
 Here we use Theorem~\ref{thm:prob-non-edge} and the bound $k = o(n)$, which follows from the fact that $p \gg n^{-1}\log n$.

Let us also recall the definition of the error terms that we use to control the deviation of the actual degrees from the expected trajectory.
For $i \ge 0$, we set
\begin{align}\label{eq:error-function}
 f_0 \coloneqq 4\log n \sqrt{ \dfrac{\log pn}{pn} + p } \qquad \text{and} \qquad f_i := \left ( \dfrac{1+16p}{1-p} \right )^i f_0.
\end{align}
Note that the error is increasing, yet we have
\begin{align}\label{eq:bound-f}
 f_{i-1} < f_{i} \le f_k \le (pn)^{2^{-5}\eps} f_0 = o(1) \qquad \text{and} \qquad
 \dfrac{1}{1\pm f_i} = 1 \pm \dfrac{3}{2}f_i
\end{align}
for all $i \in [k]$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
To derive the inequality $f_k \le (pn)^{2^{-5}\eps}f_0$, we simply use that $(1+16p)(1-p)^{-1} \le 1+2^5p \le e^{2^5p}$.
It follows from~\ref{item:P1} that
\begin{align}\label{eq:initial-parameters-2}
 |V_i| = (1\pm f_i)(1-p)^i n .
\end{align}
for all $i \in [k]$.
Other useful estimates are
\begin{align}\label{eq:asymptotics}
 (1-3p)^k \ge e^{-8pk} \ge (np)^{-2^{-7}\eps} \ge n^{-2^{-7}(\eps - \eps^2)} \ge n^{-2^{-7}\eps } \quad
 \text{and} \quad f_i > f_0 \ge n^{-1/2}
 \end{align}
for all $i \in [k]$.
Above we use that $1-x \ge e^{-2x}$ for all $x \in (0,1/2)$.

By Freedman's inequality, and using that $f_0 = o(1)$ and $\td_0 = pn$, we have
 \begin{align}\label{eq:freedman-appl-1}
 \Pr{ X_{v,i}^{-}-X_{v,0}^{-} \ge \dfrac{f_0\td_0}{2}} & \le
 \exp\left(-\frac{(f_0\td_0)^2}{2^{12}(p^2n+\log n)(pn + f_0\td_0)}\right) \nonumber \\
 & \le \exp\left(-\frac{f_0^2pn}{2^{13}(p^2n+\log n)}\right) \nonumber \\
 & \le \exp\left(-\frac{\log^2 n \log pn + p^2n \log^2 n }{2^{9}(p^2n+\log n)}\right).
 \end{align}
 In the last inequality, we use the definition of $f_0$ (see~\eqref{eq:error-function}).
 Let $E$ denote the absolute value of the exponent in the last term.
 If $p^2n \ge \log n$, then $E \ge 2^{-9}p^2n\log^2n(p^2n+\log n)^{-1} \ge 2^{-10}\log^2n$, and hence
 the right-hand side of~\eqref{eq:freedman-appl-1} is at most $\exp(-2^{-10}\log^2 n)$; if $p^2n < \log n$, then $E \ge 2^{-9}\log^2 n \log pn(p^2n+\log n)^{-1} \ge 2^{-10}\log n \log pn$, and hence
 the right-hand side of~\eqref{eq:freedman-appl-1} is at most $\exp(-2^{-10}\log n \log pn)$.
 Therefore, in both cases we have
 \begin{align*} \Pr{X_{v,i}^{-}-X_{v,0}^{-} \ge \dfrac{f_0\td_0}{2}} \le n^{-2^{-10}\log pn} \quad \text{and} \quad
 \Pr{-X_{v,i}^{+}+X_{v,0}^{+} \ge \dfrac{f_0\td_0}{2}} \le n^{-2^{-10}\log pn}.
 \end{align*}
 Summing over all $v \in V_0$ and $i \in [k-1]$, we obtain that $\Pr{\tau < k} \le n^{-2^{-11}\log pn}.$
\end{proof}

In particular, by combining our result with the lower bound in~\eqref{eq:lower-bound-MRSS}, we obtain that for all $\varepsilon > 0$ and $p$ such that $ n^{-1+\varepsilon} \le p \le n^{-\varepsilon}$, with high probability the Prague dimension of $G_{n,p}$ is $\Theta_{\varepsilon}(pn)$.
As it is crucial for our analysis to have concentration on the degrees of all vertices in $G_{n,p}$, we note that $n^{-1}\log n$ is a natural barrier for the range of $p$ our method can handle.

To prove Theorem~\ref{thm:main-prague}, instead of working directly with the Prague dimension, we use~\eqref{eq:pdim-idim} and upper bound $\mathrm{idim}(G_{n,p})$.
Our main idea is to select independent sets with size proportional to $\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen.
We then control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.

Our strategy is inspired by the work of Bennett and Bohman~\cite{Bennett2016-di}, who use a random greedy independent set process to lower bound the independence number of regular hypergraphs under certain degree and codegree conditions.
For graphs, we also improve on their work by handling pseudorandom graphs with logarithmic degrees rather than regular graphs with degrees growing polynomially with the number of vertices.
Our analysis is done via the differential equation method, which was popularised in the combinatorial community by Wormald~\cite{Wormald-1,Wormald-2}.
As can be seen in the references given in~\cite{Wormald-2}, the roots of the method may be traced back further, see for example the works of Kurtz~\cite{Kurtz1970-aa}
and Karp and Sipser~\cite{Karp1981-aa}.

Our methods also allow us to bound the minimum number of independent sets needed to cover all the non-edges of $G=G_{n,p}$, denoted by $\overline{\theta}_1(G)$.
Initially, one may conjecture that with high probability $\overline{\theta}_1(G)$ should be of order $n^2/\alpha(G)^2$  for all values of $p$, which corresponds to the size of a `near' optimal clique cover of $\overline{G}$.
Frieze and Reed~\cite{Frieze1995-xa}, by improving the results of Bollobás, Erd\H{o}s, Spencer, and West~\cite{Bollobas1993-nl}, showed that this is indeed the case for constant values of $p$.
Surprisingly, this is not true for the sparse range.
In~\cite{Guo2023-wq}, Guo, Patton and Warnke proved that for any fixed $\varepsilon >0$ and $n^{-1} \ll p \le 1-\varepsilon$, with high probability we have
\begin{align*}
	\overline{\theta}_1(G_{n,p}) = \Omega \left ( \dfrac{n^2 \log \frac{1}{p} }{\alpha^2(G_{n,p})} \right ).
\end{align*}
It is worth mentioning that due to the work of Matula~\cite{Matula1970-ka}, Bollobás and Erd\H{o}s~\cite{Bollobas1976-al}, Grimmett and McDiarmid~\cite{Grimmett1975-vs}, and Frieze~\cite{Frieze1990-cu}, we know that the size of the largest independent set in $G_{n,p}$ is of order $\Theta(p^{-1}\log pn)$ for $n^{-1} \ll p \le 1- \varepsilon$, for any fixed $\varepsilon > 0$.
Also note that
$\log \frac{1}{p} = \Theta_{\varepsilon}(\log n)$ for $ n^{-1}\ll p \ll n^{-\varepsilon}$.

Recently, Molnar, Rödl, Sales and Schacht~\cite{MolnarRodlSalesSchacht} obtained a matching upper bound for every $\varepsilon > 0$ and $p$ such that $n^{-1/2} \log^{5/4} n \ll p \ll n^{-\varepsilon}$.
Their argument heavily relies on the fact that the number of independent sets in $G_{n,p}$ is concentrated around its expectation with high probability.
As this no longer holds for $p \ll n^{-1/2}$ (see Corollary~19 in~\cite{Coja-Efthymiou}), there is a barrier at $n^{-1/2}$ for the range of $p$ their method can handle.
Our second main theorem overcomes this natural barrier and extends their result to a much wide range of~$p$.

\begin{theorem}\label{thm:main}
	Let $\varepsilon\in (0,1)$ and $(\log n)^{2+\varepsilon} n^{-1} \le p < n^{-\varepsilon}$.
	Then, with high probability we have
	\begin{align*}
		\overline{\theta}_1(G_{n,p}) = \Theta_{\varepsilon} \left ( \dfrac{n^2\log n}{\alpha^2(G_{n,p})} \right ).
	\end{align*}
\end{theorem}

It is worth noting we were only able to overcome the previous obstacles because for many host graphs the independent set generated by our random greedy process is not uniformly distributed.
As an example, let $1 \ll k \ll n$ and take a complete bipartite graph $K[A,B]$ with $|A| = n/3$ and $|B| = 2n/3$.
Now let $I$ be a uniformly-chosen random independent $k$-set and let $J$ be the independent set generated by our random greedy process.
One can check that if $u, v \in A$, then $\mathbb{P}\left[u,v \in I\right] \ll \mathbb{P}\left[u,v \in J\right]$.

Now let us mention some of the historic background related to $\overline{\theta}_1$.
In the 1960s, Erd\H{o}s, Goodman and Pósa~\cite{Erdos1966-so} introduced the parameter $\theta_1(G)$, defined as the minimum number of cliques required to cover all the edges of $G$.
Notice that $\overline{\theta}_1(G) = \theta_1(\overline{G})$, and hence one may think of $\theta_1(G)$ and $\overline{\theta}_1(G)$ as the dual versions of each other.
In~\cite{Erdos1966-so}, the authors showed that $\theta_1(G) \le \lfloor n^2/4 \rfloor$ for every $n$-vertex graph $G$, which is attained by balanced complete bipartite graphs.
In 1994, Alon~\cite{Alon1986-oc} showed that if $G$ is a $n$-vertex graph with maximum degree $\Delta$, then $\theta_1(\overline{G}) = O(\Delta^2 \log n)$.
Eaton and Rödl~\cite{Eaton1996-ap} showed that this bound is close to best possible, as there are $n$-vertex graphs $G$ with $\theta_1(\overline{G}) = \Omega(\Delta^2 \log n / \log \Delta)$.

Frieze and Reed~\cite{Frieze1995-xa} showed that with high probability $\theta_1(G_{n,p}) = \Theta_p(n^2/\log n)$ for constant values of $p$.
Guo, Patton and Warnke~\cite{Guo2023-wq} extended these results much further by showing that for every fixed $\gamma \in (0,1)$ and $p$ such that $n^{-2} \le p \le 1-\gamma$, with high probability, we have $\theta_1(G_{n,p}) = \Theta (n^2p /\log_{1/p}^{2} n )$.
As results in~\cite{Guo2023-wq} left the range where $p$ tends to 1 open, it is more convenient to just consider $\overline{\theta}_1(G_{n,p})$ for $p$ tending to 0.

The rest of the paper is divided as follows.
In Section \ref{sec:proofs}, we present the proofs of our main theorems; in Section~\ref{sec:random-greedy}, we analyse our random greedy independent set process using the differential equation method; in Section~\ref{sec:prob-cover}, we prove the main technical result of the paper, namely Theorem~\ref{thm:main-dem}; and in Section \ref{sec:preliminaries}, we present some tools and standard lemmas that will be used in the analysis of our random process.

To prove Theorem~\ref{thm:main-prague}, the authors say they do not work directly with the Prague dimension; instead they “use~\eqref{eq:pdim-idim} and upper bound $\mathrm{idim}(G_{n,p})$.” The “main idea” is to “select independent sets with size proportional to $\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen,” and then “control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.” They note the strategy is inspired by Bennett--Bohman’s random greedy independent set process, and that their “analysis is done via the differential equation method.”

To prove the main theorem, the authors say they do not work directly with the Prague dimension; instead they “use
\begin{align}\label{eq:pdim-idim}
	\idim(G) \le \pdim(G) \le \idim(G) + 1.
\end{align}
and upper bound $\mathrm{idim}(G_{n,p})$.” The “main idea” is to “select independent sets with size proportional to $\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen,” and then “control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.” They note the strategy is inspired by Bennett--Bohman’s random greedy independent set process, and that their “analysis is done via the differential equation method.”

\label{thm:main-prague}
	Let\footnote{In all our statements, $\varepsilon$ is a fixed constant and $n$ tends to infinity.} $\varepsilon \in (0,1)$ and $(\log n)^{2+\varepsilon} n^{-1} \le p < n^{-\varepsilon}$.
	Then, with high probability we have
	\begin{align*}
		\mathrm{pdim}(G_{n,p}) = O_{\varepsilon} \left ( \dfrac{pn \log n}{\log pn} \right ).
	\end{align*},

Question: Let $G_{n,p}$ be the Erd\H{o}s--R\'enyi random graph on $n$ vertices, where each edge is present independently with probability $p$. For a graph $G$, its Prague dimension $\mathrm{pdim}(G)$ is the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph. Fix $\varepsilon \in (0,1)$, assume $n \to \infty$, and suppose the edge probability satisfies $(\log n)^{2+\varepsilon} n^{-1} \le p < n^{-\varepsilon}$. Which quantitative estimate holds with high probability (that is, with probability tending to $1$ as $n\to\infty$)?

With high probability, \[\mathrm{pdim}(G_{n,p}) = O_{\varepsilon}\!\left(\frac{pn\log n}{\log(pn)}\right).\] Equivalently, there exists a constant $C_{\varepsilon}>0$ depending only on $\varepsilon$ such that, with high probability, \[\mathrm{pdim}(G_{n,p}) \le C_{\varepsilon}\frac{pn\log n}{\log(pn)}.\]

Choices:
B. With high probability, \[\mathrm{pdim}(G_{n,p}) = O_{\varepsilon}\!\left(\frac{pn\log n}{\log(1/p)}\right).\] Equivalently, there exists a constant $C_{\varepsilon}>0$ depending only on $\varepsilon$ such that, with high probability, \[\mathrm{pdim}(G_{n,p}) \le C_{\varepsilon}\frac{pn\log n}{\log(1/p)}.\]
C. With high probability, \[\mathrm{pdim}(G_{n,p}) = O_{\varepsilon}(pn\log n).\] Equivalently, there exists a constant $C_{\varepsilon}>0$ depending only on $\varepsilon$ such that, with high probability, \[\mathrm{pdim}(G_{n,p}) \le C_{\varepsilon}\,pn\log n.\]
D. With high probability, \[\mathrm{pdim}(G_{n,p}) = O\!\left(\frac{pn\log n}{\log(pn)}\right).\] Equivalently, there exists an absolute constant $C>0$ independent of $\varepsilon$ such that, with high probability, \[\mathrm{pdim}(G_{n,p}) \le C\frac{pn\log n}{\log(pn)}.\]
E. With high probability, \[\mathrm{pdim}(G_{n,p}) = \Theta_{\varepsilon}\!\left(\frac{pn\log n}{\log(pn)}\right).\] Equivalently, there exist constants $c_{\varepsilon},C_{\varepsilon}>0$ depending only on $\varepsilon$ such that, with high probability, \[c_{\varepsilon}\frac{pn\log n}{\log(pn)}\le \mathrm{pdim}(G_{n,p}) \le C_{\varepsilon}\frac{pn\log n}{\log(pn)}.\]

logarithmic denominator from process length/coverage

dropped the improving factor \(\log(pn)\) in the denominator

dependence of the implied constant on \(\varepsilon\) and the admissible \(p\)-range

upper bound upgraded to matching lower bound without the separate lower-bound argument


\label{future directions}
In this section, we provide several research questions that allow for potential extensions of this work.

First, we note that our error bounds rely on the assumption that $n

\label{prelim}
Throughout this paper, we will let $q$ be an odd prime power with prime divisor $p$. We will use $\mathcal{M}(n)$ to denote the set of polynomials of degree $n$ in $\mathbb{F}_q[x]$, an

The study of irreducible polynomials with prescribed coefficients over finite fields carries a rich history, with much progress having been made over the last three decades. While the problem of counting irreducible polynomials over finite fields can be traced back to the early 19th century when Gauss proved that there are approximately $q^n/n$ irreducible polynomials of degree $n$ in $\mathbb{F}_q[x]$, this problem with the additional restriction of prescribed coefficients was called into question in 1992 by Hansen and Mullen \cite{Hansen92}, who conjectured that for all prime powers $q \ge 3$ and positive integers $n$, one can find a monic irreducible polynomial of degree $n$ in $\mathbb{F}_q[x]$ with any single coefficient prescribed. In 1997, Wan \cite{Wan97} proved the Hansen-Mullen conjecture for $n \ge 36$ or $q > 19$, leaving finitely many cases remaining; the conjecture was proved in its entirety by Ham and Mullen, who computationally verified the remaining cases.

In general, error bounds on the number of irreducible polynomials with prescribed coefficients have played a key role in the literature of the subject. Their practical usage is mostly present in polynomial existence proving, as proving existence analytically is often much easier compared to direct approaches. However, their ability to represent bias towards some coefficient selections over others is also of interest; in the broader context of arithmetic geometry, for instance, the \textit{analytic rank} is defined to measure the bias of tensors, and has been a topic of interest to many researchers in the field \cite{moshkovitz2024quasilinearrelationpartitionanalytic}.

Since it is hard to understand why singular coefficient prescriptions differ in representation among irreducible polynomials, we are motivated to consider counting irreducible polynomials $a_0+a_1x+\dots+a_{n-1}x^{n-1}+x^n \in \mathbb{F}_q[x]$ whose coefficient vector $(a_0, \dots, a_{n-1}, 1)$ lies in a fixed affine set $V$ of points in $\mathbb{F}_q^{n+1}$, which is the objective of this paper. Indeed, such restrictions have been studied in depth in the context of positive integers and prescribed digits, primarily using tools from analytic number theory. Nevertheless, it is notable that this has also been studied briefly in the finite fields setting, which is often easier to work in than the positive integers; see \cite{Gao2021} for self-reciprocal irreducible polynomials, for example. An important work in this line of research is that of Mérai, who gave error bounds \cite{Mérai2025} for the distribution of the Rudin-Shapiro function over irreducible polynomials in $\mathbb{F}_q[x]$. This was one of the first error bounds obtained on the distribution of a polynomial function in the coefficients of irreducible polynomials over finite fields.

The goal of this paper is as follows.
\begin{quote}
\textsl{Fix polynomials $R_1, \dots, R_m$ on the coefficients of monic polynomials in $\mathbb{F}_q[x]$ of degree $n$. Find error bounds for the number $I_n(R_1, \dots, R_m)$ of monic irreducible polynomials $f$ of degree $n$ such that $(R_1(f), \dots, R_m(f)) = (0, \dots, 0)$.}
\end{quote}
Since on average, the polynomials $R_1$, \dots, $R_m$ take on equidistributed values, we should expect to have \[ I_n(R_1, \dots, R_m) \approx \frac{I(n)}{q^m}, \] where $I(n) = q^n/n + O(q^{n/2}/n)$ is the number of irreducible polynomials of degree $n$ over $\mathbb{F}_q$. Thus, we provide an upper bound on the difference $\left|I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m}\right|$.

\begin{lemma}\label{Vaughan's identity}
 Let $\Psi: \mathbb{F}_q[x] \to \mathbb{C}$ be a function with $|\Psi(f)| \le 1$. If $u$ and $v$ are two integers in $[1, n]$ with $u+v>n$, then \[ \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\Psi(f)\right| \ll n\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\Sigma_2^{1/2}, \] with \[ \Sigma_1 = \sum_{\deg g \le u+v} \left|\sum_{\deg h = n - \deg g} \Psi(gh)\right| \] and \[ \Sigma_2 = \max_{v \le i \le n-u} \max_{\deg g_1 = n-i} \sum_{\deg g_2 = n-i} \left|\sum_{\deg h = i} \Psi(hg_1)\overline{\Psi(hg_2)}\right|, \] where both sums above are taken over monic polynomials.
\end{lemma}
In particular, it suffices to obtain upper bounds on $\Sigma_1$ and $\Sigma_2$. Using polarization, we will reduce the computation to computing character sums on multilinear polynomials.

\begin{lemma}\label{Polarization result}
 Let $P_1(f), \dots, P_m(f)$ be polynomials in the coefficients of a $k$-degree polynomial $f$. Let $j=\max(\deg P_1, \dots, \deg P_m)$ and $S$ denote the set of indices $i \in [1, m]$ for which $\deg P_i = j$. Then we have \[ \left|\sum_{\deg f = k} \prod_{i=1}^m \psi_i(P_i(f))\right| \le \left(q^{k}\left|\sum_{x \in (\mathbb{F}_q^k)^j} \prod_{i \in S} \psi_i(P_i^{\circ}(x)) \right|\right)^{1/2^j},\] where the sum above is over monic $f$.
\end{lemma}
\begin{proof}
 Note that \begin{align*}
 \left|\sum_{\deg f = k} \prod_{i=1}^m \psi_i(P_i(f))\right|^2 & = \sum_{\deg f = k} \sum_{\deg h < k} \prod_{i=1}^m \psi_i(P_i(f+h))\overline{\psi_i(P_i(f))} \\ &= \left|\sum_{\deg f = k} \sum_{\deg h < k} \prod_{i=1}^m \psi_i(\Delta_h P_i(f))\right| \\ &\le \sum_{\deg f = k}\left| \sum_{\deg h < k} \prod_{i=1}^m \psi_i(\Delta_h P_i(f))\right|.
 \end{align*}
 Repeatedly squaring in this fashion, we obtain that at the $j$th squaring, \[ \left|\sum_{\deg f = k} \prod_{i=1}^m \psi_i(P_i(f))\right|^{2^j} \le q^{k} \left|\sum_{h_1, \dots, h_j} \prod_{i \in S} \psi_i(P_i^{\circ}(h_1, \dots, h_j)) \right|, \] as desired.
\end{proof}
Recall that the purpose of this application of polarization is to obtain upper bounds on the sums $\Sigma_1$ and $\Sigma_2$ from Lemma \ref{Vaughan's identity} upon selecting $\Psi(f) = \prod_{i=1}^m \psi_i(R_i(f))$. Indeed, we may consider the following choices of $P_i$, which are both polynomials in the coefficients of their arguments.
\begin{itemize}
 \item For $\Sigma_1$, let $(P_i)_g(f)=R_i(fg)$ for all $f \in \mathcal{M}(n-\deg g)$ and $1 \le i \le m$.
 \item For $\Sigma_2$, let $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$ for all $f \in \mathcal{M}(n - \max(\deg g_1, \deg g_2))$ and $1 \le i \le m$.
\end{itemize}

\subsection{Exponential sums of multilinear polynomials}
Note that Lemma \ref{Polarization result} requires us to only work with exponential sums of multilinear polynomials going forward. In order to obtain useful bounds on the exponential sums of multilinear polynomials, we require a corollary of a recent result on the comparison between the analytic rank and partition rank of tensors.
\begin{defn}
 The \emph{analytic rank} $\operatorname{AR}(T)$ of $T$ is defined as \[ \operatorname{AR}(T) := -\log_q\left(\frac{1}{q^n} \sum_{(x_1, \dots, x_n) \in \mathbb{F}_q^n} \psi_0(T(x_1, \dots, x_n))\right). \] The \emph{partition rank} $\operatorname{PR}(T)$ of $T$ is defined as the smallest number $r$ such that $T$ can be written as the sum of $r$ reducible homogeneous polynomials.
\end{defn}

\begin{lemma}\label{char sums and ranks}
For all polynomials $f$ of degree $n<p$, $\operatorname{PR}(f^{\circ}) \ge \operatorname{rank}(f)$. Consequently, for any tuple $(P_1, \dots, P_m)$ of polynomials and nontrivial tuple $(\psi_1, \dots, \psi_m)$ of additive characters on $\mathbb{F}_q$, there exists a constant $c>0$ dependent only on $\max(\deg P_1, \dots, \deg P_m)$ such that \[ \left|\sum_{x \in (\mathbb{F}_q^k)^j} \prod_{i \in S} \psi_i(P_i^{\circ}(x)) \right| \le q^{n-c \cdot \operatorname{rank}((P_i)_{i \in S})}, \] where $j = \max(\deg P_1, \dots, \deg P_m)<p$ and $S$ is the set of indices $1 \le i \le m$ for which $\deg P_i = j$.
\end{lemma}
\begin{proof}
Note that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\frac{1}{n!}f^{\circ}(x, \dots x)$. Thus, \[ \operatorname{rank}(f) = \operatorname{rank}(f^{\circ}(x, \dots x)) \le \operatorname{PR}(f^{\circ}), \] proving the first part. Let $L$ denote the set of nontrivial $\mathbb{F}_q$-combinations of $\{P_i: i \in S\}$. Observe that \begin{align*} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \prod_{i \in S} \psi_i(P_i^{\circ}(x)) \right| = \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(\sum_{i \in S} \varphi(\psi_i)P_i^{\circ}(x)\right) \right| \le \max_{P \in L} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(P^{\circ}(x)\right) \right| \end{align*} and \begin{align*} \max_{P \in L} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(P^{\circ}(x)\right) \right| &\le \max_{P \in L} q^{n-\operatorname{AR}(P^{\circ})} \\ &\le \max_{P \in L} q^{n-c \cdot \operatorname{PR}(P^{\circ})} \\ &\le \max_{P \in L} q^{n-c \cdot \operatorname{rank}(P)} \\ &= q^{n-c \cdot \operatorname{rank}((P_i)_{i \in S})}, \end{align*} which proves the second part.
\end{proof}

\section{Proof of main result}\label{main proof}
We are ready to prove our main result.
\begin{proof}[Proof of Theorem \ref{main theorem}]
By the orthogonality of characters, we have \[ I_n(R_1, \dots, R_m) = \frac{1}{q^m} \sum_{(\psi_1, \dots, \psi_m)} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)). \] Thus, \[ I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m} = \frac{1}{q^m} \sum_{(\psi_1, \dots, \psi_m) \neq \mathbf{0}} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)). \] We may rewrite the above as \begin{align*} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)) &= \sum_{f \in \mathcal{M}(n)} \mathbf{1}_{f \in \mathcal{P}(n)}\prod_{i=1}^m \psi_i(R_i(f)) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O\left(\sum_{f \in \mathcal{M}(n)} \mathbf{1}_{\Lambda(f) > 1}\right) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O(I(n)-I(n/2)) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O(q^{n/2}/n) \end{align*} Therefore, \[ \left|I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m}\right| \le \sum_{(\psi_1, \dots, \psi_m) \neq \mathbf{0}} \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\prod_{i=1}^{m} \psi_i(R_i(f))\right| \] by the triangle inequality. By Lemma \ref{Vaughan's identity}, we have for all nontrivial tuples $(\psi_1, \dots, \psi_m)$ of additive characters on $\mathbb{F}_q$ that
\[ \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\prod_{i=1}^m \psi_i(R_i(f))\right| \ll n\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\Sigma_2^{1/2} \]
for all choices of $u$ and $v$ with $u+v<n$, where
\[ \Sigma_1 = \sum_{\deg g \le u+v} \left|\sum_{\deg h = n - \deg g} \prod_{i=1}^m \psi_i((P_i)_g(h)) \right| \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} \left|\sum_{\deg h = k} \prod_{i=1}^m \psi_i((P_i)_{g_1, g_2}(h)) \right|. \]
By Lemma \ref{Polarization result}, we have
\[ \Sigma_1 \le \sum_{0 \le d \le u+v} q^{\frac{n-d}{2^j}} \sum_{\deg g = d} \left|\sum_{x \in (\mathbb{F}_q^d)^{n-d}} \prod_{i \in S} \psi_i((P_i)_g^{\circ}(x))\right|^{1/2^j} \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{\frac{k}{2^j}}\left|\sum_{x \in (\mathbb{F}_q^{n-k})^k} \prod_{i \in S} \psi_i((P_i)_{g_1, g_2}^{\circ}(x))\right|^{1/2^j}. \]
Thus, by Lemma \ref{char sums and ranks}, we have
\[ \Sigma_1 = \sum_{0 \le d \le u+v} \sum_{\deg g = d} q^{(n-d)+\frac{n-d}{2^j}-c \cdot \operatorname{rank}(((P_i)_g)_{i \in S})} \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{k-c \cdot \operatorname{rank}(((P_i)_{g_1, g_2})_{i \in S})}. \] The desired result follows.
\end{proof}

The structure of this paper is as follows. We begin by stating the preliminaries assumed throughout this paper in Section \ref{prelim}. Next, in Section \ref{lemmas}, we state and prove the lemmas that play a key role in the proof of our main result. Our approach, based on character sums, uses a function field analogue of Vaughan's identity, polarization to reduce the problem to working with multilinear exponential sums, and partition rank to bound the multilinear exponential sums. In Section \ref{main proof}, we prove the main result by applying the lemmas from Section \ref{lemmas}, and we end with Section \ref{future directions}, in which we discuss future research directions.

To prove Theorem~\ref{main theorem}, the paper’s approach is “based on character sums” and uses “a function field analogue of Vaughan's identity,” then “polarization to reduce the problem to working with multilinear exponential sums,” and finally “partition rank to bound the multilinear exponential sums.” The main result is then proved “by applying the lemmas from Section~\ref{lemmas}” in Section~\ref{main proof}.

To prove the main theorem, the paper’s approach is “based on character sums” and uses “a function field analogue of Vaughan's identity,” then “polarization to reduce the problem to working with multilinear exponential sums,” and finally “partition rank to bound the multilinear exponential sums.” The main result is then proved “by applying the lemmas from \label{lemmas}
\subsection{Vaughan's identity in $\mathbb{F}_q[x]$}
In \cite{Mérai2025}, the following variant of Vaughan's identity for arithmetic functions on $\mathbb{N}$ is utilized. This bound en” in \begin{proof}
Note that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\frac{1}{n!}f^{\circ}(x, \dots x)$. Thus, \[ \operatorname{rank}(f) = \operatorname{rank}(f^{\circ}(x, \dots x)) \le \operatorname{PR}(f^{\circ}), \] proving the first part. Let $L$ denote the set of nontrivial $\mathbb{F}_q$-combinations of $\{P_i: i \in S\}$. Observe that \begin{align*} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \prod_{i \in S} \psi_i(P_i^{\circ}(x)) \right| = \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(\sum_{i \in S} \varphi(\psi_i)P_i^{\circ}(x)\right) \right| \le \max_{P \in L} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(P^{\circ}(x)\right) \right| \end{align*} and \begin{align*} \max_{P \in L} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(P^{\circ}(x)\right) \right| &\le \max_{P \in L} q^{n-\operatorname{AR}(P^{\circ})} \\ &\le \max_{P \in L} q^{n-c \cdot \operatorname{PR}(P^{\circ})} \\ &\le \max_{P \in L} q^{n-c \cdot \operatorname{rank}(P)} \\ &= q^{n-c \cdot \operatorname{rank}((P_i)_{i \in S})}, \end{align*} which proves the second part.
\end{proof}

\section{Proof of main result}\label{main proof}
We are ready to prove our main result.
\begin{proof}[Proof of Theorem \ref{main theorem}]
By the orthogonality of characters, we have \[ I_n(R_1, \dots, R_m) = \frac{1}{q^m} \sum_{(\psi_1, \dots, \psi_m)} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)). \] Thus, \[ I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m} = \frac{1}{q^m} \sum_{(\psi_1, \dots, \psi_m) \neq \mathbf{0}} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)). \] We may rewrite the above as \begin{align*} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)) &= \sum_{f \in \mathcal{M}(n)} \mathbf{1}_{f \in \mathcal{P}(n)}\prod_{i=1}^m \psi_i(R_i(f)) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O\left(\sum_{f \in \mathcal{M}(n)} \mathbf{1}_{\Lambda(f) > 1}\right) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O(I(n)-I(n/2))  \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O(q^{n/2}/n) \end{align*} Therefore, \[ \left|I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m}\right| \le \sum_{(\psi_1, \dots, \psi_m) \neq \mathbf{0}} \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\prod_{i=1}^{m}  \psi_i(R_i(f))\right| \] by the triangle inequality. By Lemma \ref{Vaughan's identity}, we have for all nontrivial tuples $(\psi_1, \dots, \psi_m)$ of additive characters on $\mathbb{F}_q$ that
\[ \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\prod_{i=1}^m \psi_i(R_i(f))\right| \ll n\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\Sigma_2^{1/2} \]
for all choices of $u$ and $v$ with $u+v<n$, where
\[ \Sigma_1 = \sum_{\deg g \le u+v} \left|\sum_{\deg h = n - \deg g} \prod_{i=1}^m \psi_i((P_i)_g(h)) \right| \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} \left|\sum_{\deg h = k} \prod_{i=1}^m \psi_i((P_i)_{g_1, g_2}(h)) \right|. \]
By Lemma \ref{Polarization result}, we have
\[ \Sigma_1 \le \sum_{0 \le d \le u+v} q^{\frac{n-d}{2^j}} \sum_{\deg g = d} \left|\sum_{x \in (\mathbb{F}_q^d)^{n-d}} \prod_{i \in S} \psi_i((P_i)_g^{\circ}(x))\right|^{1/2^j} \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{\frac{k}{2^j}}\left|\sum_{x \in (\mathbb{F}_q^{n-k})^k} \prod_{i \in S} \psi_i((P_i)_{g_1, g_2}^{\circ}(x))\right|^{1/2^j}. \]
Thus, by Lemma \ref{char sums and ranks}, we have
\[ \Sigma_1 = \sum_{0 \le d \le u+v} \sum_{\deg g = d} q^{(n-d)+\frac{n-d}{2^j}-c \cdot \operatorname{rank}(((P_i)_g)_{i \in S})} \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{k-c \cdot \operatorname{rank}(((P_i)_{g_1, g_2})_{i \in S})}. \] The desired result follows.
\end{proof}
”.

\label{main theorem}
  Let $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q$ odd. Suppose that $n < p$. Let $j:=\max(\deg R_1, \dots, \deg R_m)$, and let $S$ denote the set of indices $1 \le i \le m$ for which $\deg R_i(f) = j$. Define $(P_i)_g(f)=R_i(fg)$ and $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$. There exists a constant $c>0$, dependent only on $j$, such that \[ \left|I_n(R_1, \dots, R_m)-\frac{I(n)}{q^m}\right| \le \frac{q^m-1}{q^m} \cdot \min_{0 \le u+v < n} \left(nS_1(u, v) + n^{5/2}q^{n-(u+v)/2}\sqrt{S_2(u, v)}\right) \] where \[ S_1(u, v) = \sum_{0 \le d \le u+v} \sum_{\deg g = d} q^{(n-d)+\frac{n-d}{2^j}-c \cdot \operatorname{rank}(((P_i)_g)_{i \in S})} \] and \[ S_2(u, v) = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{k-c \cdot \operatorname{rank}(((P_i)_{g_1, g_2})_{i \in S})}. \],

Existential–Universal
Inequality or Bound

Question:
Let \(\mathbb{F}_q\) be a finite field of odd order \(q\) and characteristic \(p\), and let \(n<p\). Let \(R_1,\dots,R_m\) be polynomial functions of the coefficients of monic degree-\(n\) polynomials over \(\mathbb{F}_q\). Define \(I_n(R_1,\dots,R_m)\) to be the number of monic irreducible polynomials \(f\in \mathbb{F}_q[x]\) of degree \(n\) such that \(R_1(f)=\cdots=R_m(f)=0\), and let \(I(n)\) denote the number of monic irreducible polynomials of degree \(n\) over \(\mathbb{F}_q\). Set \(j:=\max(\deg R_1,\dots,\deg R_m)\), let \(S=\{1\le i\le m: \deg R_i=j\}\), and define \((P_i)_g(f):=R_i(fg)\) and \((P_i)_{g_1,g_2}(f):=R_i(g_1f)-R_i(g_2f)\). For nonnegative integers \(u,v\) with \(u+v<n\), define
\[
S_1(u,v)=\sum_{0\le d\le u+v}\sum_{\deg g=d} q^{(n-d)+\frac{n-d}{2^j}-c\,\operatorname{rank}(((P_i)_g)_{i\in S})}
\]
and
\[
S_2(u,v)=\max_{v\le k\le n-u}\max_{\deg g_1=n-k}\sum_{\deg g_2=n-k} q^{k-c\,\operatorname{rank}(((P_i)_{g_1,g_2})_{i\in S})},
\]
where the degree-constrained sums are over monic polynomials. Under these assumptions, which quantitative estimate holds?

There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]

Choices:
B.
There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v\le n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]
C.
There exists a constant \(c>0\), depending only on \(j\), such that for every pair of nonnegative integers \(u,v\) with \(u+v<n\),
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]
D.
There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}S_2(u,v)\right).
\]
E.
There exists a constant \(c>0\), depending only on \(j,m\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]


Question: Let \(\mathbb{F}_q\) be a finite field of odd order \(q\) and characteristic \(p\), let \(n<p\), and let \(R_1,\dots,R_m\) be polynomials in the coefficients of monic degree-\(n\) polynomials over \(\mathbb{F}_q\). Define \(I_n(R_1,\dots,R_m)\) to be the number of monic irreducible polynomials \(f\in \mathbb{F}_q[x]\) of degree \(n\) such that \(R_1(f)=\cdots=R_m(f)=0\), and let \(I(n)\) denote the number of irreducible polynomials of degree \(n\) over \(\mathbb{F}_q\). Set \(j=\max(\deg R_1,\dots,\deg R_m)\), let \(S=\{1\le i\le m:\deg R_i=j\}\), and define \((P_i)_g(f)=R_i(fg)\) and \((P_i)_{g_1,g_2}(f)=R_i(g_1f)-R_i(g_2f)\). Which explicit uniform estimate is guaranteed?

There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right),
\]
where
\[
S_1(u,v)=\sum_{0\le d\le u+v}\ \sum_{\deg g=d} q^{(n-d)+\frac{n-d}{2^j}-c\cdot \operatorname{rank}(((P_i)_g)_{i\in S})}
\]
and
\[
S_2(u,v)=\max_{v\le k\le n-u}\ \max_{\deg g_1=n-k}\ \sum_{\deg g_2=n-k} q^{k-c\cdot \operatorname{rank}(((P_i)_{g_1,g_2})_{i\in S})}.
\]

Choices:
B.
There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v\le n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right),
\]
where
\[
S_1(u,v)=\sum_{0\le d\le u+v}\ \sum_{\deg g=d} q^{(n-d)+\frac{n-d}{2^j}-c\cdot \operatorname{rank}(((P_i)_g)_{i\in S})}
\]
and
\[
S_2(u,v)=\max_{v\le k\le n-u}\ \max_{\deg g_1=n-k}\ \sum_{\deg g_2=n-k} q^{k-c\cdot \operatorname{rank}(((P_i)_{g_1,g_2})_{i\in S})}.
\]
C.
There exists a constant \(c>0\), depending only on \(j\), such that for every pair of integers \(u,v\) with \(0\le u+v<n\),
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right),
\]
where
\[
S_1(u,v)=\sum_{0\le d\le u+v}\ \sum_{\deg g=d} q^{(n-d)+\frac{n-d}{2^j}-c\cdot \operatorname{rank}(((P_i)_g)_{i\in S})}
\]
and
\[
S_2(u,v)=\max_{v\le k\le n-u}\ \max_{\deg g_1=n-k}\ \sum_{\deg g_2=n-k} q^{k-c\cdot \operatorname{rank}(((P_i)_{g_1,g_2})_{i\in S})}.
\]
D.
There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right),
\]
where
\[
S_1(u,v)=\sum_{0\le d\le u+v}\ \sum_{\deg g=d} q^{(n-d)+\frac{n-d}{2^j}-c\cdot \operatorname{rank}((P_i)_g: i\in S)}
\]
and
\[
S_2(u,v)=\max_{v\le k\le n-u}\ \max_{\deg g_1=n-k}\ \sum_{\deg g_2=n-k} q^{k-c\cdot \operatorname{rank}((P_i)_{g_1,g_2}: i=1,\dots,m)}.
\]
E.
There exists a constant \(c=c(j,q)>0\), depending only on \(j\) and \(q\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}S_2(u,v)\right),
\]
where
\[
S_1(u,v)=\sum_{0\le d\le u+v}\ \sum_{\deg g=d} q^{(n-d)+\frac{n-d}{2^j}-c\cdot \operatorname{rank}(((P_i)_g)_{i\in S})}
\]
and
\[
S_2(u,v)=\max_{v\le k\le n-u}\ \max_{\deg g_1=n-k}\ \sum_{\deg g_2=n-k} q^{k-c\cdot \operatorname{rank}(((P_i)_{g_1,g_2})_{i\in S})}.
\]

dropped the outer minimization over \((u,v)\), keeping a bound valid for each admissible pair

rank taken on the wrong family of polynomials, replacing the top-degree index set \(S\) by all indices in \(S_2\)

loss of the square-root in the \(\Sigma_2\) term and altered dependence of the constant