\begin{observation}\label{Chain}
For any graph $G$ with maximum degree $\Delta$ and positive integer $k\in[\Delta]$,
\begin{center}
$\chi(G)=\chi_{\Delta}^{f}(G)\leq \ldots\leq \chi_{2}^{f}(G)\leq \chi_{1}^{f}(G)=\chi(G^{2})$.
\end{center}
\end{observation}

A proper vertex coloring of a graph $G$ is $t$-\textit{frugal} if no color appears more than $t$ times in the neighborhood of any vertex in $G$. The $t$-\textit{frugal chromatic number} of $G$ is the minimum $k$ for which $G$ admits a $t$-frugal coloring with $k$ colors, and we denote it by $\chi_{t}^{f}(G)$. Hind, Molloy and Reed~\cite{HMR} initiated the concept of frugal coloring in 1997 proving that a graph $G$ with sufficiently large maximum degree $\Delta$ admits a $({\log}^8{\Delta})$-frugal coloring with $\Delta+1$ colors and mentioning an application for total colorings. The result was improved by Ndreca, Procacci and Scoppola~\cite{nps} in 2012, while only a few other authors considered this type of coloring so far~\cite{AEv,BMR,KM}; see also~\cite{ch-2011} where its list version was considered. In this paper, we expand the consideration of frugal colorings by studying several aspects that are of relevance for coloring invariants.

Frugal colorings are related to several known coloring invariants. A {\em $2$-distance coloring} of a graph $G$ is a mapping $c:V(G)\to \{1,\ldots,k\}$ such that any two vertices at distance at most $2$ receive different colors. The minimum number of colors in such a coloring is the {\em $2$-distance chromatic number} $\chi_{2}(G)$ of $G$; see Kramer and Kramer~\cite{kk0} for systematic treatment of $d$-distance coloring. Alternatively, $2$-distance coloring of $G$ coincides with the coloring of the square $G^2$ of $G$, where two vertices in $G^2$ are adjacent if they are adjacent or have a common neighbor in $G$, and so $\chi_{2}(G)=\chi(G^{2})$. Clearly, the condition of $2$-frugal coloring is weaker than that of $2$-distance coloring, thus $\chi_{2}^{f}(G)\le \chi_2(G)$ for any graph $G$. A proper coloring of a graph $G$ in which the vertices of any two color classes induce a forest of paths is a {\em linear coloring} of $G$, as introduced by Yuster~\cite{yus}. Note that a linear coloring is a $2$-frugal coloring of $G$, but the converse is not necessarily true. Thus, for the resulting invariant ${\rm lc}(G)$ of $G$, which is the minimum number of colors in a linear coloring in $G$, we get $\chi_{2}^{f}(G)\le {\rm lc}(G)$.

Another concept related to frugal coloring arises from that of limited packing as introduced by Gallant et al.~\cite{GGHR} and studied further in~\cite{GZ,BS}. A set $B\subseteq V(G)$ is a {\em $t$-limited packing} if the closed neighborhood of each vertex in $G$ contains at most $t$ vertices in $B$. Note that when $t=1$, the resulting $1$-limited packing coincides with the concept of $2$-packing, as introduced by Meir and Moon~\cite{MM} and studied in many papers concerning graph domination. Recently, a \textit{$t$-limited packing partition} of $G$, which is a partition of $V(G)$ into $t$-limited packings, was considered and the following coloring invariant was introduced.
The \textit{$t$-limited packing partition number}, denoted by $\chi_{\times t}(G)$, is the minimum cardinality of a $t$-limited packing partition in $G$; see~\cite{AS,AL}.
Note that when $t\ge 2$, the partition is not necessarily a proper coloring. However, additionally imposing that the color classes in a $t$-limited packing partition are independent results precisely in a $t$-frugal coloring. In particular, $\chi_{t}^{f}(G)\ge \chi_{\times t}(G)$ holds for any graph $G$ and any positive integer $t$.

The behavior of a particular graph coloring is inherently related to the nature and structure of its color classes. On the other hand, the color classes of many types of graph colorings have been studied independently. For example, a large number of papers have been published about ``independent", ``$2$-packing", ``open packing" and ``dissociation" sets, which are color classes of the ``standard", ``$2$-distance", ``injective" and ``defective" colorings, respectively. Moreover, the color classes of frugal colorings are a variant of limited packings, notably, they are independent limited packings. Due to the fundamental role of these sets in the study of frugal colorings, we investigate them under the name of {\em frugal independent sets}. More specifically, such a set is a {\em $t$-frugal independent set} ($t$FI set for short), where $t$ is a positive integer. The maximum cardinality of a $t$FI set in $G$ will be denoted by $\alpha_t^f(G)$, and called $t$-{\em frugal independence number} of $G$.
In line with the above arguments, we will investigate frugal independent sets along with frugal colorings both from computational and combinatorial points of view.

\subsection{Preliminaries}

Throughout the paper, we consider $G$ as a simple graph with vertex set $V(G)$ and edge set $E(G)$. In addition, $G$ is finite unless explicitly stated otherwise. We use~\cite{West} as a reference for terminology and notation which are not explicitly defined here. The (\textit{open}) {\em neighborhood} of a vertex $v$ is denoted by $N_{G}(v)$, and its {\em closed neighborhood} is $N_{G}[v]=N_{G}(v)\cup \{v\}$. The {\em minimum} and {\em maximum degrees} of $G$ are denoted by $\delta(G)$ and $\Delta(G)$, respectively. Given subsets $A,B\subseteq V(G)$, let $[A,B]$ denote the set of all edges with one endvertex in $A$ and the other in $B$. For simplicity, we use the notation $[k]$ instead of $\{1,\ldots,k\}$ for any positive integer $k$. By a $\chi_{t}^{f}(G)$-coloring we mean a $t$-frugal coloring of $G$ with $\chi_{t}^{f}(G)$ colors.

Since $\chi_{t}^{f}(G)=\chi(G)$ for each integer $t\ge \Delta(G)$, where $\chi(G)$ is the chromatic number of $G$,  we restrict our attention to the cases when $t\in[\Delta(G)-1]$. Moreover, the following inequality chain follows from the definitions.

A proper vertex coloring of a graph $G$ is $t$-\textit{frugal} if no color appears more than $t$ times in the neighborhood of any vertex in $G$. The $t$-\textit{frugal chromatic number} of $G$ is the minimum $k$ for which $G$ admits a $t$-frugal coloring with $k$ colors, and we denote it by $\chi_{t}^{f}(G)$. Hind, Molloy and Reed~\cite{HMR} initiated the concept of frugal coloring in 1997 proving that a graph $G$ with sufficiently large maximum degree $\Delta$ admits a $({\log}^8{\Delta})$-frugal coloring with $\Delta+1$ colors and mentioning an application for total colorings. The result was improved by Ndreca, Procacci and Scoppola~\cite{nps} in 2012, while only a few other authors considered this type of coloring so far~\cite{AEv,BMR,KM}; see also~\cite{ch-2011} where its list version was considered. In this paper, we expand the consideration of frugal colorings by studying several aspects that are of relevance for coloring invariants.

Frugal colorings are related to several known coloring invariants. A {\em $2$-distance coloring} of a graph $G$ is a mapping $c:V(G)\to \{1,\ldots,k\}$ such that any two vertices at distance at most $2$ receive different colors. The minimum number of colors in such a coloring is the {\em $2$-distance chromatic number} $\chi_{2}(G)$ of $G$; see Kramer and Kramer~\cite{kk0} for systematic treatment of $d$-distance coloring. Alternatively, $2$-distance coloring of $G$ coincides with the coloring of the square $G^2$ of $G$, where two vertices in $G^2$ are adjacent if they are adjacent or have a common neighbor in $G$, and so $\chi_{2}(G)=\chi(G^{2})$. Clearly, the condition of $2$-frugal coloring is weaker than that of $2$-distance coloring, thus $\chi_{2}^{f}(G)\le \chi_2(G)$ for any graph $G$. A proper coloring of a graph $G$ in which the vertices of any two color classes induce a forest of paths is a {\em linear coloring} of $G$, as introduced by Yuster~\cite{yus}. Note that a linear coloring is a $2$-frugal coloring of $G$, but the converse is not necessarily true. Thus, for the resulting invariant ${\rm lc}(G)$ of $G$, which is the minimum number of colors in a linear coloring in $G$, we get $\chi_{2}^{f}(G)\le {\rm lc}(G)$.

Another concept related to frugal coloring arises from that of limited packing as introduced by Gallant et al.~\cite{GGHR} and studied further in~\cite{GZ,BS}. A set $B\subseteq V(G)$ is a {\em $t$-limited packing} if the closed neighborhood of each vertex in $G$ contains at most $t$ vertices in $B$. Note that when $t=1$, the resulting $1$-limited packing coincides with the concept of $2$-packing, as introduced by Meir and Moon~\cite{MM} and studied in many papers concerning graph domination. Recently, a \textit{$t$-limited packing partition} of $G$, which is a partition of $V(G)$ into $t$-limited packings, was considered and the following coloring invariant was introduced.
The \textit{$t$-limited packing partition number}, denoted by $\chi_{\times t}(G)$, is the minimum cardinality of a $t$-limited packing partition in $G$; see~\cite{AS,AL}.
Note that when $t\ge 2$, the partition is not necessarily a proper coloring. However, additionally imposing that the color classes in a $t$-limited packing partition are independent results precisely in a $t$-frugal coloring. In particular, $\chi_{t}^{f}(G)\ge \chi_{\times t}(G)$ holds for any graph $G$ and any positive integer $t$.

The behavior of a particular graph coloring is inherently related to the nature and structure of its color classes. On the other hand, the color classes of many types of graph colorings have been studied independently. For example, a large number of papers have been published about ``independent", ``$2$-packing", ``open packing" and ``dissociation" sets, which are color classes of the ``standard", ``$2$-distance", ``injective" and ``defective" colorings, respectively. Moreover, the color classes of frugal colorings are a variant of limited packings, notably, they are independent limited packings. Due to the fundamental role of these sets in the study of frugal colorings, we investigate them under the name of {\em frugal independent sets}. More specifically, such a set is a {\em $t$-frugal independent set} ($t$FI set for short), where $t$ is a positive integer. The maximum cardinality of a $t$FI set in $G$ will be denoted by $\alpha_t^f(G)$, and called $t$-{\em frugal independence number} of $G$.
In line with the above arguments, we will investigate frugal independent sets along with frugal colorings both from computational and combinatorial points of view.

Throughout the paper, we consider $G$ as a simple graph with vertex set $V(G)$ and edge set $E(G)$. In addition, $G$ is finite unless explicitly stated otherwise. We use~\cite{West} as a reference for terminology and notation which are not explicitly defined here. The (\textit{open}) {\em neighborhood} of a vertex $v$ is denoted by $N_{G}(v)$, and its {\em closed neighborhood} is $N_{G}[v]=N_{G}(v)\cup \{v\}$. The {\em minimum} and {\em maximum degrees} of $G$ are denoted by $\delta(G)$ and $\Delta(G)$, respectively. Given subsets $A,B\subseteq V(G)$, let $[A,B]$ denote the set of all edges with one endvertex in $A$ and the other in $B$. For simplicity, we use the notation $[k]$ instead of $\{1,\ldots,k\}$ for any positive integer $k$. By a $\chi_{t}^{f}(G)$-coloring we mean a $t$-frugal coloring of $G$ with $\chi_{t}^{f}(G)$ colors.

Since $\chi_{t}^{f}(G)=\chi(G)$ for each integer $t\ge \Delta(G)$, where $\chi(G)$ is the chromatic number of $G$, we restrict our attention to the cases when $t\in[\Delta(G)-1]$. Moreover, the following inequality chain follows from the definitions.

Since $\chi_{t}^{f}(G)=\chi(G)$ for each integer $t\ge \Delta(G)$, where $\chi(G)$ is the chromatic number of $G$, we restrict our attention to the cases when $t\in[\Delta(G)-1]$. Moreover, the following inequality chain follows from the definitions.

Let $\mathcal{J}$ be the color classes of a $\chi_{t}^{f}(G)$-coloring, and let $u$ be a vertex in $G$ of maximum degree. Let $u\in J$, where $J\in \mathcal{J}$. By definition, $J$ is an independent set and $u$ has at most $t$ neighbors in each color class in $\mathcal{J}\setminus \{J\}$. This shows that
\begin{equation}\label{Delta}
\Delta(G)=\deg_{G}(u)=\sum_{J'\in \mathcal{J}\setminus \{J\}}|N_{G}(u)\cap J'|\leq t|\mathcal{J}\setminus \{J\}|=t\fk(G)-t.
\end{equation}
Therefore, $\fk(G)\geq \lceil \Delta(G)/t\rceil+1$. The following lower bound on the $t$-frugal chromatic number follows directly from the latter inequality and Observation \ref{Chain}.

\begin{observation}
\label{ob:boundDelta}
If $G$ is a graph with maximum degree $\Delta$, then $\chi_{t}^{f}(G)\le 1+\Delta(1+\big \lfloor\frac{\Delta-1}{t}\big\rfloor)$.
\end{observation}
The bound in Observation~\ref{ob:boundDelta} can be sharp. For instance, when $t=2$ and $\Delta=2$, the bound reads $\chi_{2}^{f}(G)\le 1+2(1+\big \lfloor\frac{2-1}{2}\big\rfloor)=3$, and $\chi_{2}^{f}(C_{2r+1})=3$ where $r\in\mathbb{N}$.

\begin{theorem}\label{General1}
If $t$ is a positive integer and $G$ is a graph of size $m$, then
\begin{center}
$\chi_{t}^{f}(G)\geq \dfrac{1}{2}+\sqrt{\dfrac{1}{4}+\dfrac{2m}{t\alpha_{t}^{f}(G)}}$
\end{center}
with equality if and only if $G\in \Psi_{t}$.
\end{theorem}
\begin{proof}
The lower bound trivially holds for edgeless graphs. So, we assume that $\chi_{t}^{f}(G)\geq2$. Let $\mathbb{B}=\{B_{1},\ldots,B_{\chi_{t}^{f}(G)}\}$ be a $\chi_{t}^{f}(G)$-coloring. Without loss of generality, we may assume that $|B_{1}|\leq \ldots\leq|B_{\chi_{t}^{f}(G)}|$. Then,
\begin{equation}\label{Ine}
\begin{array}{lcl}
m&=&\sum_{s=1}^{\chi_{t}^{f}(G)-1}\sum_{t=s+1}^{\chi_{t}^{f}(G)}|[B_{s},B_{t}]|\leq t\sum_{s=1}^{\chi_{t}^{f}(G)-1}|B_{s}|(\chi_{t}^{f}(G)-s)\vspace{1mm}\\
&\leq& t\alpha_{t}^{f}(G)\sum_{s=1}^{\chi_{t}^{f}(G)-1}(\chi_{t}^{f}(G)-s)=t\alpha_{t}^{f}(G)\big{(}\dfrac{\chi_{t}^{f}(G)(\chi_{t}^{f}(G)-1)}{2}\big{)}.
\end{array}
\end{equation}
Therefore, $t\alpha_{t}^{f}(G)\chi_{t}^{f}(G)^{2}-t\alpha_{t}^{f}(G)\chi_{t}^{f}(G)-2m\geq0$. Solving this inequality for $\chi_{t}^{f}(G)$, we obtain the desired lower bound on $\chi_{t}^{f}(G)$.

Now, let us prove that any graph $G\in \Psi_{t}$ attains the lower bound. Let $X_{1},\ldots,X_{r}$ be the partite sets of $G$. For any two distinct indices $i,j\in[r]$, $|X_{i}|=|X_{j}|$ follows from the fact that every vertex in $X_{i}$ (resp. $X_{j}$) has precisely $t$ neighbors in $X_{j}$ (resp. $X_{i}$).
By the structure of $G$, each partite set $X_{t}$ is a $t$FI set. This shows that $|X_{t}|\leq \alpha_{t}^{f}(G)$ and that $\chi_{t}^{f}(G)\leq r$. Let $B$ be an $\alpha_{t}^{f}(G)$-set. Suppose to the contrary that $|X_{t}|<\alpha_{t}^{f}(G)$. This in particular implies that $B\nsubseteq X_{t}$. Suppose that $X_{t}\subseteq B$. In such a situation, the strict inequality $|X_{t}|<\alpha_{t}^{f}(G)$ shows that $B$ contains a vertex $v$ from a partite set $X_{j}$ with $j\neq t$. This contradicts the fact that $B$ is an independent set as $v$ has a neighbor in $X_{t}$ by the structure of $G$. Therefore, $X_{t}\nsubseteq B$. Now set $Q=\{(b,x):\, b\in B\setminus X_{t}, x\in X_{t}\setminus B\ \mbox{and}\ bx\in E(G)\}$. Since $B$ is a $t$FI set, every vertex in $X_{t}\setminus B$ is adjacent to at most $t$ vertices in $B\setminus X_{t}$. So, $|Q|\leq t|X_{t}\setminus B|=t|X_{t}|-t|B\cap X_{t}|$. On the other hand, every vertex in $B\setminus X_{t}$ has exactly $t$ neighbors in $X_{t}\setminus B$ by the structure of $G$ and since $B$ is an independent set. This shows that $|Q|=t|B\setminus X_{t}|=t|B|-t|B\cap X_{t}|$. This, together with the last inequality, results in $|B|\leq|X_{t}|$. Therefore, $\alpha_{t}^{f}(G)=|X_{1}|=\ldots=|X_{r}|=|V(G)|/r$. With this in mind, we have
\begin{center}
$\chi_{t}^{f}(G)\geq \dfrac{1}{2}+\sqrt{\dfrac{1}{4}+\dfrac{2m}{t\alpha_{t}^{f}(G)}}=\dfrac{1}{2}+\sqrt{\dfrac{1}{4}+r^{2}-r}=r$.
\end{center}
This leads to the desired equality.

\begin{proposition}\label{TF}
For any triangle-free graph $G$ of order $n$,
\begin{center}
$\chi_{2}^{f}(G)\leq \Big{\lfloor}\dfrac{n-\alpha_2^{f}(G)+4}{2}\Big{\rfloor}$
\end{center}
and this bound is sharp.
\end{proposition}
\begin{proof}
If $n\leq \alpha_2^{f}(G)+2$, then the upper bound is easily verified. So, we may assume that $n\geq \alpha_2^{f}(G)+3$. Let $B$ be an $\alpha_2^{f}(G)$-set and set $G'=G[V(G)\setminus B]$. Clearly, $G'$ is a triangle-free graph as well. With this in mind and since $|V(G')|\geq3$, once can partition $V(G')$ into $t$ subsets $B_{1},\ldots,B_{t}$ in such a way that
\begin{itemize}
 \item $B_{i}$ consists of two nonadjacent vertices for each $i\in[t-1]$, and
 \item $|B_{t}|\in \{1,2\}$.
\end{itemize}
If $|B_{t}|=1$ or $B_{t}$ consists of two nonadjacent vertices, then $\{B,B_{1},\ldots,B_{t}\}$ is a $2$-frugal coloring of $G$. Therefore, $\chi_{2}^{f}(G)\leq2+(n-\alpha_2^{f}(G)-1)/2=(n-\alpha_2^{f}(G)+3)/2$ or $\chi_{2}^{f}(G)\leq1+\big{(}n-\alpha_2^{f}(G)\big{)}/2=(n-\alpha_2^{f}(G)+2)/2$ if $|B_{t}|=1$ or $|B_{t}|=2$, respectively.

\begin{theorem}\label{Lexi}
For any graphs $G$ and $H$,
\begin{center}
$\chi_{2}^{f}(H)+\Big{\lceil}\frac{\Delta(G)|V(H)|}{2}\Big{\rceil}\leq \chi_{2}^{f}(G\circ H)\leq \chi_{2}^{f}(G)|V(H)|$.
\end{center}
These bounds are sharp.
\end{theorem}
\begin{proof}
Let $\{B_{1},\ldots,B_{\chi_{2}^{f}(G)}\}$ be a $\chi_{2}^{f}(G)$-coloring and let $V(H)=\{h_{1},\ldots,h_{|V(H)|}\}$. Let $f:V(G\circ H)\to [\chi_{2}^{f}(G)|V(H)|]$ be defined by
$f(g,h)=(t-1)n+j$, where $g\in B_t$ and $h=h_j$.
Note that $B_{t}\times \{h_{j}\}$, for $t\in[\chi_{2}^{f}(G)]$ and $j\in[|V(H)|]$, are the color classes of $f$.
Since $B_{t}$ is an I$2$F set in $G$, the adjacency role of the lexicographic product graphs shows that every color class $B_{t}\times \{h_{j}\}$ is an independent set in $G\circ H$. Suppose that there exists a vertex $(g,h)\in V(G\circ H)$ adjacent to three vertices $(g_{1},h_{j})$, $(g_{2},h_{j})$ and $(g_{3},h_{j})$ in $B_{t}\times \{h_j\}$ for some $t\in[\chi_{2}^{f}(G)]$ and $j\in[|V(H)|]$. This in particular implies that $g_{1},g_{2},g_{3}\in N_{G}[g]$. Since $B_{t}$ is an $2$FI set in $G$, we may assume that $g=g_{1}$ and $g_{2},g_{3}\in N_{G}(g)$. This is impossible because $B_{t}$ is an independent set in $G$. Therefore, $B_{t}\times \{h_{j}\}$ is an I$2$F set in $G\circ H$ for each $t\in[\chi_{2}^{f}(G)]$ and $j\in[|V(H)|]$. Thus, $\chi_{2}^{f}(G\circ H)\leq|f\big{(}V(G\circ H)\big{)}|=\chi_{2}^{f}(G)|V(H)|$.

Let $\mathcal{J}$ be the color classes of a $\chi_{t}^{f}(G)$-coloring, and let $u$ be a vertex in $G$ of maximum degree. Let $u\in J$, where $J\in \mathcal{J}$. By definition, $J$ is an independent set and $u$ has at most $t$ neighbors in each color class in $\mathcal{J}\setminus \{J\}$. This shows that
\begin{equation}\label{Delta}
\Delta(G)=\deg_{G}(u)=\sum_{J'\in \mathcal{J}\setminus \{J\}}|N_{G}(u)\cap J'|\leq t|\mathcal{J}\setminus \{J\}|=t\chi_{t}^{f}(G)-t.
\end{equation}
Therefore, $\chi_{t}^{f}(G)\geq \lceil \Delta(G)/t\rceil+1$. The following lower bound on the $t$-frugal chromatic number follows directly from the latter inequality and Observation \ref{Chain}.

\begin{observation}
\label{ob:lowerboundfor2}
If $G$ is a graph and $t\ge1$, then $\chi_{t}^{f}(G)\ge \max\{\chi(G),\big\lceil\frac{\Delta(G)}{t}\big\rceil+1\}.$
\end{observation}

Concerning upper bounds with respect to the maximum degree of a graph, we obtain the following observation, which immediately follows by using a greedy coloring algorithm.

\begin{observation}
\label{ob:boundDelta}
If $G$ is a graph with maximum degree $\Delta$, then $\chi_{t}^{f}(G)\le 1+\Delta(1+\big \lfloor\frac{\Delta-1}{t}\big\rfloor)$.
\end{observation}
The bound in Observation~\ref{ob:boundDelta} can be sharp. For instance, when $t=2$ and $\Delta=2$, the bound reads $\chi_{2}^{f}(G)\le 1+2(1+\big \lfloor\frac{2-1}{2}\big\rfloor)=3$, and $\chi_{2}^{f}(C_{2r+1})=3$ where $r\in\mathbb{N}$.

\subsection{Main results and organization of the paper}

We start with computational aspects of the two main invariants studied in this paper. As proved in~\cite{BMR}, the decision version of the $t$-frugal chromatic number is NP-complete for any positive integer $t$. Hence, in Section~\ref{sec:computational}, we concentrate on computational aspects with respect to the $t$-frugal independence number and prove that the decision version of $\alpha_t^f(G)$ is NP-complete for any positive integer $t$ even if restricted to bipartite graphs $G$. In contrast, we present a linear-time algorithm for computing $\alpha_t^f(T)$ in an arbitrary tree $T$ and for any $t$.

In Section~\ref{sec:generalbounds}, we prove several general bounds involving one or both graph invariants. In particular, we prove that $\chi_{t}^{f}(G)\geq {1}/{2}+\sqrt{{1}/{4}+{2m}/({t\alpha_{t}^{f}(G))}}$ holds for any positive integer $t$ and any graph $G$ of size $m$, and we characterize the graphs that attain this bound. On the other hand, when $G$ is triangle-free, we prove an upper bound on $\chi_{2}^{f}(G)$ expressed in terms of the order of $G$ and its $2$-frugal independence number. In addition, we prove an upper bound on the $2$-frugal independence number of an arbitrary graph $G$ expressed in terms of the order of $G$ and several other invariants that depend on its pendant vertices. In Section~\ref{sec:subcubic} we focus on subcubic graphs, where our main result is that any graph $G$ with $\Delta(G)=3$ satisfies $3\le \chi_{2}^{f}(G)\le 5$. While the lower bound is trivial, the upper bound is proved by an extensive case analysis. If, in addition, $G$ is claw-free, then the equality $\chi_{2}^{f}(G)=3$ holds.

Section~\ref{sec:NG} is devoted to Nordhaus-Gaddum results, where we again focus on the $t$-frugal chromatic number where $t=2$. We prove that
$$\dfrac{n}{2}+2\le\chi_{2}^{f}(G)+\chi_{2}^{f}(\overline{G})\leq \dfrac{3n}{2},$$
where the lower bound holds for all graphs $G$ of order $n$ with the exception of $6$ sporadic graphs, while the upper holds for all graphs $G$. In addition, the upper bound holds with equality if and only if $n$ is even and $G\in \{K_{1,n-1},\overline{K_{1,n-1}}\}$.

In Section~\ref{sec:classes}, we consider several graph classes and graph operations. For an arbitrary block graph $G$ we obtain the exact value of the $2$-frugal chromatic number, that is,
 $\chi_{2}^{f}(G)=\max\big\{\omega,\big\lceil\frac{\Delta}{2}\big\rceil+1\big\}$, where $\omega$ is the clique number and $\Delta$ the maximum degree of $G$. If $G\,\square\, H$ is the Cartesian product of graphs $G$ and $H$ we obtain the following lower and upper bound on its $2$-frugal chromatic number: $\max\{\chi_{2}^{f}(G),\chi_{2}^{f}(H)\}\le \chi_{2}^{f}(G\,\square\, H)\le \max\{\chi_2(G),\chi_2(H)\}$. The bounds are sharp and in some cases coincide. We also obtain sharp upper bounds for the $2$-frugal chromatic numbers in strong and direct products of two graphs. In addition, we bound the lexicographic product of two graphs as follows: $\chi_{2}^{f}(H)+\Big{\lceil}\frac{\Delta(G)|V(H)|}{2}\Big{\rceil}\leq \chi_{2}^{f}(G\circ H)\leq \chi_{2}^{f}(G)|V(H)|$, and provide examples of sharpness. We continue with the quest for classes of graphs that achieve the trivial lower bound for their $2$-frugal chromatic number, notably when $\chi_{2}^{f}(G)=\lceil\Delta(G)/2\rceil+1$. We prove that several well-known infinite lattices satisfy this equality. In addition, the equality holds for Cartesian products of several (infinite) paths as well as strong products of several (infinite) paths. We conclude the paper with some open problems and directions for future research.

Let $\mathcal{J}$ be the color classes of a $\chi_t^f(G)$-coloring, and let $u$ be a vertex of maximum degree with $u\in J\in\mathcal{J}$. Since $J$ is independent and $u$ has at most $t$ neighbors in each other color class $J'\in\mathcal{J}\setminus\{J\}$, one has
\[
\Delta(G)=\deg_G(u)=\sum_{J'\in\mathcal{J}\setminus\{J\}} |N_G(u)\cap J'|\le t\,|\mathcal{J}\setminus\{J\}|=t\chi_t^f(G)-t.
\]
Therefore $\chi_t^f(G)\ge \lceil \Delta(G)/t\rceil+1$. The stated lower bound $\chi_t^f(G)\ge \max\{\chi(G),\lceil \Delta(G)/t\rceil+1\}$ is said to follow directly from this inequality together with Observation~\ref{Chain} (the chain $\chi(G)=\chi_\Delta^f(G)\le\cdots\le\chi_1^f(G)=\chi(G^2)$).

For an upper bound in terms of $\Delta$, it is stated that one "immediately" obtains $\chi_t^f(G)\le 1+\Delta\bigl(1+\lfloor(\Delta-1)/t\rfloor\bigr)$ by using a greedy coloring algorithm.

Let $\mathcal{J}$ be the color classes of a $\chi_t^f(G)$-coloring, and let $u$ be a vertex of maximum degree with $u\in J\in\mathcal{J}$. Since $J$ is independent and $u$ has at most $t$ neighbors in each other color class $J'\in\mathcal{J}\setminus\{J\}$, one has
\[
\Delta(G)=\deg_G(u)=\sum_{J'\in\mathcal{J}\setminus\{J\}} |N_G(u)\cap J'|\le t\,|\mathcal{J}\setminus\{J\}|=t\chi_t^f(G)-t.
\]
Therefore $\chi_t^f(G)\ge \lceil \Delta(G)/t\rceil+1$. The stated lower bound $\chi_t^f(G)\ge \max\{\chi(G),\lceil \Delta(G)/t\rceil+1\}$ is said to follow directly from this inequality together with the relevant monotonicity chain used in establishing the main theorem (namely that $\chi_t^f(G)$ is at least $\chi(G)$).

For an upper bound in terms of $\Delta$, it is stated that one "immediately" obtains $\chi_t^f(G)\le 1+\Delta\bigl(1+\lfloor(\Delta-1)/t\rfloor\bigr)$ by using a greedy coloring algorithm.,

Question: Let $G$ be a finite simple graph with maximum degree $\Delta$. For a positive integer $t$, a proper vertex coloring of $G$ is called $t$-frugal if no color appears more than $t$ times in the open neighborhood of any vertex of $G$, and $\chi_t^f(G)$ denotes the minimum number of colors in a $t$-frugal coloring of $G$. Also, $G^2$ denotes the square of $G$, whose vertices are those of $G$ and where two vertices are adjacent whenever they are at distance at most $2$ in $G$, so $\chi(G^2)$ is the $2$-distance chromatic number of $G$. Which statement holds for every such graph $G$?

$\chi(G)=\chi_{\Delta}^{f}(G)\le \chi_{\Delta-1}^{f}(G)\le \cdots \le \chi_{2}^{f}(G)\le \chi_{1}^{f}(G)=\chi(G^{2}).$

Choices:
B. $\chi(G)=\chi_{\Delta}^{f}(G)\le \chi_{\Delta-1}^{f}(G)\le \cdots \le \chi_{2}^{f}(G)\le \chi_{1}^{f}(G)\le \chi(G^{2}).$
C. $\chi(G)\le \chi_{\Delta}^{f}(G)\le \chi_{\Delta-1}^{f}(G)\le \cdots \le \chi_{2}^{f}(G)\le \chi_{1}^{f}(G)=\chi(G^{2}).$
D. $\chi(G)=\chi_{\Delta}^{f}(G)\ge \chi_{\Delta-1}^{f}(G)\ge \cdots \ge \chi_{2}^{f}(G)\ge \chi_{1}^{f}(G)=\chi(G^{2}).$
E. $\chi(G)=\chi_{\Delta}^{f}(G)\le \chi_{\Delta-1}^{f}(G)\le \cdots \le \chi_{2}^{f}(G)=\chi_{1}^{f}(G)=\chi(G^{2}).$

endpoint equality at t=1 replaced by mere upper bound

dropped the sharp equality \chi(G)=\chi_\Delta^f(G) at the left endpoint

strictness/equality at the t=1 and t=2 end of the chain

The stem gives only definitions of t-frugal coloring, G^2, and the relevant invariants; it does not reveal the claimed monotonicity direction or the endpoint equalities. There is no direct answer leakage.

The item is close to asking for a standard theorem/proposition verbatim: monotonicity of χ_t^f in t together with the endpoint identities χ_Δ^f(G)=χ(G) and χ_1^f(G)=χ(G^2). It is not pure restatement because the choices vary the direction/strength of the claim, but it still mainly tests recall of the exact statement.

Some reasoning is needed to see why larger t weakens the constraint, why t=Δ recovers ordinary coloring, and why t=1 corresponds to coloring G^2. However, the task is undermined by the presence of choice C, which is also true as a weaker consequence, so the item does not cleanly force selection of a unique strongest conclusion.

Several distractors are mathematically plausible and target common errors: reversing monotonicity, incorrectly making inequalities strict, and confusing χ(G^2) with χ_2(G). But choice C is a genuinely true statement, not a distractor, which creates ambiguity and substantially weakens distractor quality.

Moderately good mathematical content, but flawed as a single-correct MCQ because it includes another true option (C). The stem avoids leakage, yet the question is close to theorem recall and does not cleanly test uniquely identifiable reasoning.


The algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs).
Historically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \textit{Notebooks}, particularly his series for $1/\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\pi^2$, characterized by a polynomial architecture with quartic partial numerators.
This identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.
In this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.

\section{Introduction}
The algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs).
Historically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \textit{Notebooks}, particularly his series for $1/\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\pi^2$, characterized by a polynomial architecture with quartic partial numerators.
This identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.
In this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.

Conjecture~\eqref{conj:main} presents a notable structural complexity where the complexity of the coefficients (quartic in $n$) seemingly contradicts the simplicity of the limit. In this paper, we provide a formal analytic regularization of this conjecture. By constructing a discrete Green's function analogue through operator factorization, we resolve the recurrence into a summation series, thereby proving the identity.

\section{Preliminaries}
The evaluation of GCFs of the form
\begin{equation}
x = b_0 + \K_{n=1}^{\infty} \left( \frac{a_n}{b_n} \right) = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \dots}}
\end{equation}
is fundamentally governed by the relationship between its partial coefficients and its convergents $A_n/B_n$.

\begin{proposition}\label{prop:factor}
For the conjectured coefficients $b_n = 3n(n+1)+1$ and $a_n = -(2n-1)n^3$, the recurrence admits an exact reduction over $\mathbb{N}^{+}$ with the auxiliary sequences:
\begin{equation}
c_n = n^2, \quad d_n = n(2n-1).
\end{equation}
\end{proposition}

\subsubsection{Series Representation of the Convergent Ratio}
The $n$-th convergent of the continued fraction, $x_n = A_n/B_n$, is formulated by the ratio of \eqref{eq:Bn_sum_final} to \eqref{eq:An_product_final}.
The common product terms cancel exactly, yielding:
\begin{equation}\label{eq:xn_final}
x_n = \frac{A_n}{B_n} = \left( \sum_{k=0}^{n} u_k \right)^{-1}, \quad \text{where } u_k = \frac{\prod_{j=1}^{k} c_j}{\prod_{j=1}^{k+1} d_j}.
\end{equation}
This fundamental equivalence maps the convergence of the polynomial continued fraction to the partial sums of an infinite series. The proof of the conjecture~\eqref{conj:main} is thereby reduced to the asymptotic evaluation of the sum $S = \sum_{k=0}^{\infty} u_k$.

The convergence of the GCF is predicated on the existence of the limit of the convergents $\{x_n\}$. According to~\eqref{eq:xn_final}, the value of the GCF, denoted by $\mathcal{K}$, is mapped to the reciprocal of an infinite series:
\begin{equation}\label{eq:limit_mapping}
\mathcal{K} = \lim_{n \to \infty} \frac{A_n}{B_n} = \left( \sum_{k=0}^{\infty} t_k \right)^{-1}, \quad \text{where } t_k = \frac{\prod_{j=1}^{k} c_j}{\prod_{j=1}^{k+1} d_j}.
\end{equation}

\begin{theorem}[Pincherle's Theorem~\cite{Gautschi1967}]
The generalized continued fraction $b_0 + \K_{n=1}^\infty (a_n / b_n)$ converges to a finite value if and only if the associated second-order linear difference equation
\begin{equation}\label{eq:rec_pincherle}
 y_n = b_n y_{n-1} + a_n y_{n-2}
\end{equation}
admits a minimal solution $\{f_n\}$. A non-trivial solution $\{f_n\}$ is minimal if, for any solution $\{g_n\}$ linearly independent of $\{f_n\}$, the following asymptotic condition is satisfied:
\begin{equation}
 \lim_{n \to \infty} \frac{f_n}{g_n} = 0.
\end{equation}
If such a solution exists, the GCF converges to $f_0/f_{-1}$.
\end{theorem}

Crucially, from the perspective of Pincherle’s Theorem, the fact that $S = \pi^2/8$ is a \textbf{finite constant} ensures that $B_n$ grows at an asymptotically faster rate than $A_n$ in the sense of the solution hierarchy. Any solution not proportional to $\{A_n\}$ will eventually be dominated by the summation-induced growth. Thus, the existence of this convergent series confirms $\{A_n\}$ as the minimal solution, and the GCF converges to the reciprocal of the sum:
\begin{equation}
 \mathcal{K} = S^{-1} = \left( \sum_{m=1}^{\infty} \frac{2^m}{m^2 \binom{2m}{m}} \right)^{-1} = \frac{8}{\pi^2}.
\end{equation}
The proof is complete.
\end{proof}

\begin{conjecture}\label{conj:main}
The constant $8/\pi^2$ admits the following GCF representation:
\begin{equation}\label{eq:conjecture}
\frac{8}{\pi^2} = 1 - \cfrac{2 \cdot 1^4 - 1^3}{7 - \cfrac{2 \cdot 2^4 - 2^3}{19 - \cfrac{2 \cdot 3^4 - 3^3}{37 - \dots}}}
\end{equation}
where the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.
\end{conjecture}

To prove Conjecture~\eqref{conj:main}, the paper "provide[s] a formal analytic regularization" of the continued fraction. The argument proceeds by "constructing a discrete Green's function analogue through operator factorization," and then using this construction to "resolve the recurrence into a summation series," which "thereby prov[es] the identity."

To prove the main conjecture, the paper "provide[s] a formal analytic regularization" of the continued fraction. The argument proceeds by "constructing a discrete Green's function analogue through operator factorization," and then using this construction to "resolve the recurrence into a summation series," which "thereby prov[es] the identity."

\label{conj:main}
The constant $8/\pi^2$ admits the following GCF representation:
\begin{equation}\label{eq:conjecture}
\frac{8}{\pi^2} = 1 - \cfrac{2 \cdot 1^4 - 1^3}{7 - \cfrac{2 \cdot 2^4 - 2^3}{19 - \cfrac{2 \cdot 3^4 - 3^3}{37 - \dots}}}
\end{equation}
where the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.,

Question:
Let
\[
1+\K_{n=1}^{\infty}\left(\frac{a_n}{b_n}\right)
\]
denote a generalized continued fraction, where
\[
a_n=-(2n^4-n^3)=-(2n-1)n^3,\qquad b_n=3n^2+3n+1=3n(n+1)+1\quad (n\ge 1).
\]
Equivalently, this is
\[
1- \cfrac{2\cdot 1^4-1^3}{7-\cfrac{2\cdot 2^4-2^3}{19-\cfrac{2\cdot 3^4-3^3}{37-\ddots}}}.
\]
Which explicit value does this generalized continued fraction represent?

It equals \(\dfrac{8}{\pi^2}\); that is,
\[
\frac{8}{\pi^2}=1- \cfrac{2\cdot 1^4-1^3}{7-\cfrac{2\cdot 2^4-2^3}{19-\cfrac{2\cdot 3^4-3^3}{37-\ddots}}}.
\]

Choices:
B.
It equals \(\dfrac{4}{\pi^2}\); that is,
\[
\frac{4}{\pi^2}=1- \cfrac{2\cdot 1^4-1^3}{7-\cfrac{2\cdot 2^4-2^3}{19-\cfrac{2\cdot 3^4-3^3}{37-\ddots}}}.
\]
C.
It equals the reciprocal of a finite positive constant; equivalently, the generalized continued fraction converges to some real number \(L>0\).
\[
1- \cfrac{2\cdot 1^4-1^3}{7-\cfrac{2\cdot 2^4-2^3}{19-\cfrac{2\cdot 3^4-3^3}{37-\ddots}}}=L.
\]
D.
It equals \(\dfrac{8}{\pi}\); that is,
\[
\frac{8}{\pi}=1- \cfrac{2\cdot 1^4-1^3}{7-\cfrac{2\cdot 2^4-2^3}{19-\cfrac{2\cdot 3^4-3^3}{37-\ddots}}}.
\]
E.
It equals \(\left(\sum_{m=1}^{\infty}\dfrac{2^m}{m^2\binom{2m}{m}}\right)^{-1}\), and this reciprocal sum is \(\dfrac{6}{\pi^2}\); that is,
\[
1- \cfrac{2\cdot 1^4-1^3}{7-\cfrac{2\cdot 2^4-2^3}{19-\cfrac{2\cdot 3^4-3^3}{37-\ddots}}}=\left(\sum_{m=1}^{\infty}\frac{2^m}{m^2\binom{2m}{m}}\right)^{-1}=\frac{6}{\pi^2}.
\]

The stem gives only the continued fraction data and asks for its value; it does not state or strongly hint at 8/pi^2. There is no explicit answer leakage.

The item is not a direct restatement of a conclusion already embedded in the stem. It asks the solver to identify an explicit evaluation among competing alternatives.

Determining the exact constant represented by this nontrivial generalized continued fraction requires substantial mathematical recognition or derivation, so the correct answer is not obvious from surface features alone.

Several distractors are plausible mathematical confusions (4/pi^2, 8/pi, and a related-looking reciprocal sum), but choice C is a weaker true statement rather than a genuinely false distractor, which weakens single-answer integrity.

A strong, non-leaky, reasoning-heavy MCQ, but its distractor set is weakened by the inclusion of a true-but-weaker option.


\begin{enumerate}
        \item There exists a valuation $v$ on $R$ centered at $\m$ with $A(v)<\infty$ such that $v(\a_j)=b_j$ for all $j\ge 1$;
        \item For every $r\ge 1$,
    \[\vb_{I_{\blt}^{(r)}}\big(\a_1\cdots\a_r\big)=\sum_{j=1}^r b_j,\]
    and it holds that
    \[\sup_{r\ge 1}\sup_{k\in\bbN}\Big(\lct^{\a_1^k\cdots\a_r^k}\big( I^{(r)}_{\blt}\big)-k\sum_{j=1}^r b_j\Big)<\infty.\]
    \end{enumerate}
    Here $I^{(r)}_{\blt}=\sum_{j=1}^{r}\big(\frac{1}{b_j}\cdot \a_j\big)_{\blt}$.
\end{theorem}

\vspace{.1in} {\bf Organization}. This paper is organized as follows. In \Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\Rightarrow$ (3) in \Cref{thm-finite.interp}. In \Cref{sec-f.v.i}, we prove \Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\QM(Y,D)$ of quasi-monomial valuations. In \Cref{sec-i.v.i}, we prove \Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \cite{BGMY25}. Finally, we demonstrate how \Cref{thm-finite.interp} recovers a result in \cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.

\vspace{.1in} {\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.

\section{Valuation and asymptotic Samuel function}\label{sec-r.o}

In this section, $R$ is a noetherian integral domain.

\subsection{Valuation}\label{subsec-val}

Recall that a function $v\colon R^*=R\setminus\{0\} \to \bbR_{\ge 0}$ is a (real) \emph{valuation} if
\[v(xy)=v(x)+v(y) \quad \& \quad v(x+y)\ge \min\{v(x),v(y)\},\]
for all $x,y\in R^*$. Set $v(0)=\infty$. A valuation $v$ on $R$ can be extended uniquely to a valuation $v\colon K\to \bbR\cup\{\infty\}$ on the field of fractions $K$ of $R$ by setting $v(x/y)=v(x)-v(y)$. Say $v\le w$ for two valuations $v,w$ on $R$ if $v(x)\le w(x)$ for all $x\in R^*$. For any nonzero ideal $\a$ of $R$, define $v(\a)=\inf_{x\in \a}v(x)$. Then it is easy to check: for nonzero ideals $\a,\b$ in $R$,
\[v(\a\b)=v(\a)+v(\b) \quad \& \quad v(\a+\b)= \min\{v(\a),v(\b)\}.\]
In particular, we can see if the ideals $\a_1,\ldots,\a_r$ in $R$ satisfies $\sum_{j=1}^r\a_j=R$, then
\[0=v(R)=v\big(\sum \a_j\big)=\min v(\a_j),\]
which implies $v(\a_j)=0$ for some $j\in\{1,\ldots,r\}$.

\subsection{Graded sequence and filtration of ideals}\label{subsec-gr.fil}

A \emph{graded sequence} of ideals $\a_{\blt}=(\a_m)_{m\in\bbZ_{>0}}$ is a sequence of ideals in $R$ that satisfies $\a_p\cdot \a_p\subseteq \a_{p+q}$ for all $p,q\ge 1$. Put $\a_0=R$. We always assume $\a_m\neq (0)$ for some $m\ge 1$. For example, $v$ is a nontrivial valuation on $R$, then $\a_{\blt}^v$ is a graded sequence of ideals in $R$ defined by $\a_{m}^v=\{x\in R \mid v(x)\ge m\}$.

Now let $v$ be a valuation on $R$, and $\a_{\blt}$ a graded sequence of ideals in $R$. Then $v(\a_{p+q})\le v(\a_p)+v(\a_q)$ for all $p,q\ge 1$, which implies the limit
\[v(\a_{\blt})\coloneqq \lim_{m\to \infty, \, \a_m\neq (0)}\frac{v(\a_m)}{m}=\inf_{m\ge 1}\frac{v(\a_m)}{m}\]
exists by Fekete's lemma. In particular, $v(\a_{\blt}^v)=1$ for any nontrivial valuation $v$ on $R$. A graded sequence $\a_{\blt}=\{\a_m\}_{m\in\bbN}$ of ideals in $R$ is called a \emph{filtration} if additionally it satisfies $\a_m\supseteq \a_{m+1}$ for every $m\in\bbN$.

\subsection{Asymptotic Samuel function}

The classical asymptotic Samuel function (\cite{Sa52}) has been generalized to the filtration case in \cite{CP24}. Let us recall the constructions and some basic properties. Let $J_{\blt}$ be a filtration of ideals in $R$. For every ideal $\a$ in $R$, the \emph{order} of $\a$ with respect to $J_{\blt}$ is defined by
\[\vJ\big(\a\big)\coloneqq \sup\,\{m\in\bbN \mid \a\subseteq J_m\}\in\bbN\cup\{\infty\}.\]
Then we can see $\vJ(\a^{p+q})\ge \vJ(\a^p)+\vJ(\a^q)$ for every $p,q\in\bbN$. Therefore, it follows that
\[\vb(\a;J_{\blt})\coloneqq \lim_{m\to\infty}\frac{\vJ(\a^m)}{m}=\sup_{m\ge 1}\frac{\vJ(\a^m)}{m}\in \bbR_{\ge 0}\cup \{\infty\}\]
is well-defined by Fekete's lemma, called the \emph{asymptotic Samuel function} of $J_{\blt}$ (cf. \cite[Theorem 3.4]{CP24}). We also denote $\vb(\a;J_{\blt})\coloneqq \vbJ(\a)$ in the present paper.

\begin{example}\label{ex-asf}
    If $J_{\blt}=\{J^m\}_{m\ge 0}$ for a nonzero ideal $J$ in $R$, then $\vb(\a;J_{\blt})=\ol{\nu}_{J}(\a)$, which is the classical asymptotic Samuel function (cf. \cite{Sa52,HS06,LJT09}).
\end{example}

\begin{example}
    If we take $J_{\blt}=\a_{\blt}^v$ the filtration of valuation ideals associated to some valuation $v$ on $R$, then clearly $\vb(\a;J_{\blt})=v(\a)$.
\end{example}

The following lemma shows some basis properties of the asymptotic Samuel function, which are actually known in \cite{CP24}. The proof below follows from the strategy of \cite[Theorem 2]{Sa52}.

\begin{lemma}[{see \cite[Proposition 3.11]{CP24}}]\label{lem-mu.subadditivity}
    Let $J_{\blt}$ be a filtration of ideals in $R$. Then
    \begin{enumerate}
    \item $\vb(\a;J_{\blt})\ge \vb(\b;J_{\blt})$, for ideals $\a\subseteq\b$ in $R$;
    \item $\vb(\a^k;J_{\blt})=k\cdot\vb(\a;J_{\blt})$, where $\a$ is an ideal in $R$ and $k\in\bbZ_+$;
    \item $\vb(\a+\b;J_{\blt})= \min\{\vb(\a;J_{\blt}), \vb(\b;J_{\blt})\}$, where $\a,\b$ are ideals in $R$;
    \item $\vb(\a\b;J_{\blt})\ge \vb(\a;J_{\blt}) +\vb(\b;J_{\blt})$, where $\a,\b$ are ideals in $R$.
    \end{enumerate}

\begin{theorem}\label{thm-finite.interp}
    The following three statements are equivalent.
    \begin{enumerate}
    \item There exists a valuation $v$ on $R$ such that $v(\a_j)=b_j$ for $j=1,\dots,r$.
    \item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\a_j)=b_j$ for $j=1,\dots,r$.
    \item The equality $\vbI(\a_1\cdots\a_r)=\sum_{j=1}^r b_j$ holds.
    \end{enumerate}
\end{theorem}

Let $R$ be an excellent regular domain of equicharacteristic $0$ and of dimension $n$.

Let $\a_1,\ldots,\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\ldots,b_r\in\mathbb{R}_{\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \cite[Theorem 1.2]{BGMY25} deals with the case where $R=\O_{\mathbb{C}^n,o}$ the ring of holomorphic germs):

\begin{question}
Can one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\a_j)=b_j$ for $j=1,\ldots, r$?
\end{question}

We introduce some notations for explaining our results in the present paper. Suppose $\mathfrak{a}$ is an ideal on $X$, and $\lambda\in\bbR_{\ge 0}\cup\{\infty\}$. Denote by $(\lambda\cdot\mathfrak{a})_{\bullet}=\big((\lambda\cdot\mathfrak{a})_m\big)_{m\in\mathbb{N}}$ the \emph{filtration} (cf. \Cref{subsec-gr.fil}) of ideals such that each term:
\begin{equation*}
    (\lambda\cdot\mathfrak{a})_m=
    \begin{cases}
        R & \lambda=0 \\
        \mathfrak{a}^{\lceil \lambda m\rceil} & 0<\lambda<\infty \\
        0 & \lambda=\infty \\
    \end{cases}
    \quad \quad m\in\mathbb{N}.
\end{equation*}

If $(\b_1)_{\bullet},\ldots,(\b_s)_{\bullet}$ are filtrations of ideals in $R$, then the summation $\mathfrak{c}_{\bullet}\coloneqq \sum(\b_i)_{\bullet}$ of these filtrations is a filtration defined as:
\begin{equation*}
        \mathfrak{c}_{m}=\sum_{\substack{m_1+\cdots+m_s=m \\ m_i\in\mathbb{N}}}(\b_1)_{m_1}\cdots(\b_s)_{m_s}, \quad m\in\mathbb{N}.
\end{equation*}

Now we return to the valuative interpolation problem. For the ideals $\a_1,\ldots,\a_r$ and non-negative real numbers $b_1,\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\sum_{b_j>0}\a_j\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\a_j)=b_j$ for all $j$ (cf. \Cref{subsec-val}).

Consider the filtration
\[I_{\bullet}\coloneqq \Big(\frac{1}{b_1}\cdot \a_{1}\Big)_{\bullet}+\cdots+\Big(\frac{1}{b_r}\cdot \a_{r}\Big)_{\bullet}.\]
Following Cutkosky and Peaharaj \cite[Definition 3.5]{CP24}, the \emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\mathfrak{a}$ of $R$, let
\[\vbI(\mathfrak{a})\coloneqq \lim_{k\to\infty}\frac{\vI\big(\mathfrak{a}^k\big)}{k}\in \bbR_{\ge 0}\cup\{\infty\},\]
where $\vI\big(\mathfrak{a}\big)\coloneqq \sup\,\{m\in\mathbb{N} \mid \mathfrak{a}\subseteq I_m\}$ denotes the \emph{order function} of $I_{\bullet}$. This construction generalizes the classical asymptotic Samuel function \cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \cite{Ras06} (cf. \Cref{sec-c.a.c}).

Let $\a_1,\ldots,\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\ldots,b_r\in\mathbb{R}_{\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \cite[Theorem 1.2]{BGMY25} deals with the case where $R=\O_{\mathbb{C}^n,o}$ the ring of holomorphic germs):

\begin{question}
Can one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\a_j)=b_j$ for $j=1,\ldots, r$?
\end{question}

We introduce some notations for explaining our results in the present paper. Suppose $\mathfrak{a}$ is an ideal on $X$, and $\lambda\in\bbR_{\ge 0}\cup\{\infty\}$. Denote by $(\lambda\cdot\mathfrak{a})_{\bullet}=\big((\lambda\cdot\mathfrak{a})_m\big)_{m\in\mathbb{N}}$ the \emph{filtration} (cf. \Cref{subsec-gr.fil}) of ideals such that each term:
\begin{equation*}
 (\lambda\cdot\mathfrak{a})_m=
 \begin{cases}
 R & \lambda=0 \\
 \mathfrak{a}^{\lceil \lambda m\rceil} & 0<\lambda<\infty \\
 0 & \lambda=\infty \\
 \end{cases}
 \quad \quad m\in\mathbb{N}.
\end{equation*}

If $(\b_1)_{\bullet},\ldots,(\b_s)_{\bullet}$ are filtrations of ideals in $R$, then the summation $\mathfrak{c}_{\bullet}\coloneqq \sum(\b_i)_{\bullet}$ of these filtrations is a filtration defined as:
\begin{equation*}
 \mathfrak{c}_{m}=\sum_{\substack{m_1+\cdots+m_s=m \\ m_i\in\mathbb{N}}}(\b_1)_{m_1}\cdots(\b_s)_{m_s}, \quad m\in\mathbb{N}.
\end{equation*}

Now we return to the valuative interpolation problem. For the ideals $\a_1,\ldots,\a_r$ and non-negative real numbers $b_1,\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\sum_{b_j>0}\a_j\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\a_j)=b_j$ for all $j$ (cf. \Cref{subsec-val}).

Consider the filtration
\[I_{\bullet}\coloneqq \Big(\frac{1}{b_1}\cdot \a_{1}\Big)_{\bullet}+\cdots+\Big(\frac{1}{b_r}\cdot \a_{r}\Big)_{\bullet}.\]
Following Cutkosky and Peaharaj \cite[Definition 3.5]{CP24}, the \emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\mathfrak{a}$ of $R$, let
\[\vbI(\mathfrak{a})\coloneqq \lim_{k\to\infty}\frac{\vI\big(\mathfrak{a}^k\big)}{k}\in \bbR_{\ge 0}\cup\{\infty\},\]
where $\vI\big(\mathfrak{a}\big)\coloneqq \sup\,\{m\in\mathbb{N} \mid \mathfrak{a}\subseteq I_m\}$ denotes the \emph{order function} of $I_{\bullet}$. This construction generalizes the classical asymptotic Samuel function \cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \cite{Ras06} (cf. \Cref{sec-c.a.c}).

Consider the filtration
\[I_{\blt}\coloneqq \Big(\frac{1}{b_1}\cdot \a_{1}\Big)_{\blt}+\cdots+\Big(\frac{1}{b_r}\cdot \a_{r}\Big)_{\blt}.\]
Following Cutkosky and Peaharaj \cite[Definition 3.5]{CP24}, the \emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\a$ of $R$, let
\[\vbI(\a)\coloneqq \lim_{k\to\infty}\frac{\vI\big(\a^k\big)}{k}\in \bbR_{\ge 0}\cup\{\infty\},\]
where $\vI\big(\a\big)\coloneqq \sup\,\{m\in\bbN \mid \a\subseteq I_m\}$ denotes the \emph{order function} of $I_{\blt}$. This construction generalizes the classical asymptotic Samuel function \cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \cite{Ras06} (cf. \Cref{sec-c.a.c}).

In fact, the equivalence of the first two statements, along with the equivalence of the existence of \emph{Zhou valuations} (cf. \cite{BGMY24,BGZ25}) satisfying the same interpolation conditions, has been proved in \cite[Appendix A.1]{BGMY25}.

\begin{theorem}\label{thm-infinite.interp}
 Additionally assume $R$ is local with the unique maximal ideal $\m$. Let $(\a_j)_{j\in\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\sqrt{\sum_{j=1}^{\infty} \a_j}=\m$, and let $(b_j)_{j\in\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:
 \begin{enumerate}
 \item There exists a valuation $v$ on $R$ centered at $\m$ with $A(v)<\infty$ such that $v(\a_j)=b_j$ for all $j\ge 1$;
 \item For every $r\ge 1$,
 \[\vb_{I_{\blt}^{(r)}}\big(\a_1\cdots\a_r\big)=\sum_{j=1}^r b_j,\]
 and it holds that
 \[\sup_{r\ge 1}\sup_{k\in\bbN}\Big(\lct^{\a_1^k\cdots\a_r^k}\big( I^{(r)}_{\blt}\big)-k\sum_{j=1}^r b_j\Big)<\infty.\]
 \end{enumerate}
 Here $I^{(r)}_{\blt}=\sum_{j=1}^{r}\big(\frac{1}{b_j}\cdot \a_j\big)_{\blt}$.
\end{theorem}

\begin{proposition}\label{prop-v(aj)=bj.implies.mu=bj-1}
 If $v$ is valuation on $R$ such that $v(\a_j)\ge b_j$ for $j=1,\ldots, r$, then $\vb(\a_1\cdots\a_r;I_{\blt})\le \sum_{j=1}^r v(\a_j)$. Moreover, if $v(\a_j) = b_j$ for all $j$, then $\vb(\a_1\cdots\a_r;I_{\blt})=\sum_{j=1}^r b_j$.
\end{proposition}

\begin{lemma}\label{lem-finite.val.interp.exist.QM}
 Let $\a_0$ be a nonzero ideal of $R$ with $\vb(\a_0;I_{\blt})>0$. Suppose $(Y,D)$ is a log smooth pair over $X$ which gives a log resolution of $\a_1\cdots\a_r$. Then there exists a quasi-monomial valuation $v\in \QM(Y,D)$ such that
 \[\vb(\a_0;I_{\blt})=\frac{v(\a_0)}{v(I_{\blt})}.\]
\end{lemma}

\begin{proposition}\label{prop-val.interp.gr.seq}
 Assume $\ord_{\xi}(J_{\blt})>0$ and $\m_{\xi}^p\subseteq J_{q}$ for some $p,q\in\bbZ_+$. Let $\a$ be a nonzero ideal of $R$ with $\a\subseteq\m_{\xi}$. Then the following two statements are equivalent:
 \begin{enumerate}
 \item There exists $v\in\Val_{X,\xi}^{<\infty}$ such that $v(J_{\blt})=1$ and $v(\a)=\rho(\a;J_{\blt})$;
 \item It holds that
 \[M\coloneqq \sup_{k\in\bbN}\Big(\lct(\a^k;J_{\blt})-\rho(\a;J_{\blt})\cdot k\Big)<\infty.\]
 \end{enumerate}
 Moreover, the valuation $v$ in (1) can be chosen with $A(v)\le M$ and $v(\m_{\xi})\ge q/p$ if (2) holds.
\end{proposition}

\begin{lemma}\label{lem-multiply.change.to.plus}
 Let $J_{\blt}$ be a filtration of nonzero ideals in $R$ with $\lct(J_{\blt})<\infty$, and let $\a_1,\ldots,\a_r$ be nonzero ideals in $R$. Set $\a=\a_1\cdots\a_r$. Suppose there is a valuation $v\in\Val_X^{<\infty}$ such that $v(\a)=\rho(\a;J_{\blt})$ and $v(J_{\blt})=1$. Then $v(\a_j)=\rho(\a_j;J_{\blt})$ for $j=1,\ldots,r$ if and only if $\rho(\a;J_{\blt})=\sum_{j=1}^r \rho(\a_j;J_{\blt})$.
\end{lemma}

\begin{proposition}\label{prop-gr.seq.infinite.compute}
 Let $J_{\blt}$ be a filtration of ideals in $R$ such that $\ord_{\xi}(J_{\blt})>0$ and $\m_{\xi}^p\subseteq J_{q}$ for some $p,q\in\bbZ_+$. Assume $(\a_j)_{j\in\bbZ_+}$ is a sequence of nonzero ideals of $R$ with $\a_j\subseteq\m_{\xi}$ for all $j\ge 1$. Then the following two statements are equivalent:
 \begin{enumerate}
 \item There exists $v\in\Val_{X,\xi}^{<\infty}$ such that $v(J_{\blt})=1$ and $v(\a_j)=\rho(\a_j;J_{\blt})$ for all $j\ge 1$;
 \item For every $r\in\bbZ_+$,
 \[\rho(\a_1\cdots\a_r;J_{\blt})=\sum_{j=1}^r\rho(\a_j;J_{\blt}),\]
 and it holds that
 \[\sup_{r\ge 1}\sup_{k\in\bbN}\Big(\lct(\a_1^k\cdots\a_r^k; J_{\blt})-\rho(\a_1\cdots\a_r;J_{\blt})\cdot k\Big)<\infty.\]
 \end{enumerate}
\end{proposition}

In fact, the equivalence of the first two statements, along with the equivalence of the existence of \emph{Zhou valuations} (cf. \cite{BGMY24,BGZ25}) satisfying the same interpolation conditions, has been proved in \cite[Appendix A.1]{BGMY25}.

In addition, for the infinite valuative interpolation problem, we apply the method of valuation approximation to prove the following result in the present paper.

\begin{theorem}\label{thm-infinite.interp}
    Additionally assume $R$ is local with the unique maximal ideal $\mathfrak{m}$. Let $(\a_j)_{j\in\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\sqrt{\sum_{j=1}^{\infty} \a_j}=\mathfrak{m}$, and let $(b_j)_{j\in\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:
    \begin{enumerate}
        \item There exists a valuation $v$ on $R$ centered at $\mathfrak{m}$ with $A(v)<\infty$ such that $v(\a_j)=b_j$ for all $j\ge 1$;
        \item For every $r\ge 1$,
    \[\vb_{I_{\bullet}^{(r)}}\big(\a_1\cdots\a_r\big)=\sum_{j=1}^r b_j,\]
    and it holds that
    \[\sup_{r\ge 1}\sup_{k\in\mathbb{N}}\Big(\lct^{\a_1^k\cdots\a_r^k}\big( I^{(r)}_{\bullet}\big)-k\sum_{j=1}^r b_j\Big)<\infty.\]
    \end{enumerate}
    Here $I^{(r)}_{\bullet}=\sum_{j=1}^{r}\big(\frac{1}{b_j}\cdot \a_j\big)_{\bullet}$.
\end{theorem}

\vspace{.1in} {\bf Organization}. This paper is organized as follows. In \Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\Rightarrow$ (3) in \Cref{thm-finite.interp}. In \Cref{sec-f.v.i}, we prove \Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\QM(Y,D)$ of quasi-monomial valuations. In \Cref{sec-i.v.i}, we prove \Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \cite{BGMY25}. Finally, we demonstrate how \Cref{thm-finite.interp} recovers a result in \cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.

\vspace{.1in} {\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.

The post-theorem introduction only indicates where proofs appear and what methods are used: it says they “quickly prove the implication (1) $\Rightarrow$ (3) in \Cref{thm-finite.interp}” in \Cref{sec-r.o}; they “prove \Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\QM(Y,D)$ of quasi-monomial valuations” in \Cref{sec-f.v.i}; and they “prove \Cref{thm-infinite.interp} using valuation approximation.” No further proof outline for \Cref{thm-finite.interp} is given beyond this organizational summary.

The post-theorem introduction only indicates where proofs appear and what methods are used: it says they “quickly prove the implication (1) $\Rightarrow$ (3)” of the main theorem later; they “prove the main theorem by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\QM(Y,D)$ of quasi-monomial valuations” later; and they “prove the following theorem using valuation approximation. \begin{theorem}\label{thm-infinite.interp}
    Additionally assume $R$ is local with the unique maximal ideal $\m$. Let $(\a_j)_{j\in\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\sqrt{\sum_{j=1}^{\infty} \a_j}=\m$, and let $(b_j)_{j\in\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:
    \begin{enumerate}
        \item There exists a valuation $v$ on $R$ centered at $\m$ with $A(v)<\infty$ such that $v(\a_j)=b_j$ for all $j\ge 1$;
        \item For every $r\ge 1$,
    \[\vb_{I_{\blt}^{(r)}}\big(\a_1\cdots\a_r\big)=\sum_{j=1}^r b_j,\]
    and it holds that
    \[\sup_{r\ge 1}\sup_{k\in\bbN}\Big(\lct^{\a_1^k\cdots\a_r^k}\big( I^{(r)}_{\blt}\big)-k\sum_{j=1}^r b_j\Big)<\infty.\]
    \end{enumerate}
    Here $I^{(r)}_{\blt}=\sum_{j=1}^{r}\big(\frac{1}{b_j}\cdot \a_j\big)_{\blt}$.
\end{theorem} No further proof outline for the main theorem is given beyond this organizational summary.

Question:
Let \(\mathfrak a_1,\ldots,\mathfrak a_r\) be nonzero ideals of a ring \(R\), and let \(b_1,\ldots,b_r\in \mathbb R_{\ge 0}\). For \(\lambda\in \mathbb R_{\ge 0}\cup\{\infty\}\) and an ideal \(\mathfrak a\), define the filtration \((\lambda\cdot \mathfrak a)_\bullet=((\lambda\cdot \mathfrak a)_m)_{m\in\mathbb N}\) by
\[
(\lambda\cdot \mathfrak a)_m=
\begin{cases}
R,& \lambda=0,\\
\mathfrak a^{\lceil \lambda m\rceil},& 0<\lambda<\infty,\\
0,& \lambda=\infty.
\end{cases}
\]
If \((\mathfrak b_1)_\bullet,\ldots,(\mathfrak b_s)_\bullet\) are filtrations, their sum is the filtration \(\mathfrak c_\bullet\) with
\[
\mathfrak c_m=\sum_{m_1+\cdots+m_s=m}(\mathfrak b_1)_{m_1}\cdots(\mathfrak b_s)_{m_s}.
\]
Set
\[
I_\bullet:=\Big(\frac1{b_1}\cdot \mathfrak a_1\Big)_\bullet+\cdots+\Big(\frac1{b_r}\cdot \mathfrak a_r\Big)_\bullet,
\]
where \(1/0\) is interpreted as \(\infty\). For any ideal \(\mathfrak a\subseteq R\), define the order function
\[
\nu_I(\mathfrak a):=\sup\{m\in\mathbb N\mid \mathfrak a\subseteq I_m\},
\]
and the associated asymptotic Samuel function
\[
\bar\nu_I(\mathfrak a):=\lim_{k\to\infty}\frac{\nu_I(\mathfrak a^k)}{k}.
\]
Which statement holds for these data?

The following three statements are equivalent: (i) there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); (ii) there exists a quasi-monomial valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); and (iii) the equality \(\bar\nu_I(\mathfrak a_1\cdots \mathfrak a_r)=\sum_{j=1}^r b_j\) holds.

Choices:
B. The following three statements are equivalent: (i) there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); (ii) there exists a quasi-monomial valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); and (iii) for every nonempty subset \(J\subseteq\{1,\dots,r\}\), one has \(\bar\nu_I\big(\prod_{j\in J}\mathfrak a_j\big)=\sum_{j\in J} b_j\).
C. If there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\), then \(\bar\nu_I(\mathfrak a_1\cdots \mathfrak a_r)=\sum_{j=1}^r b_j\).
D. The following three statements are equivalent: (i) there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); (ii) there exists a quasi-monomial valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); and (iii) for every ideal \(\mathfrak a\subseteq R\), one has \(\bar\nu_I(\mathfrak a)=\inf_{v}\frac{v(\mathfrak a)}{v(I_\bullet)}\), where the infimum runs over all valuations \(v\) on \(R\) with \(v(I_\bullet)>0\).
E. The following three statements are equivalent: (i) there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); (ii) there exists a quasi-monomial valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); and (iii) the inequality \(\bar\nu_I(\mathfrak a_1\cdots \mathfrak a_r)\ge \sum_{j=1}^r b_j\) holds.

single-product equality upgraded to all partial products

dropped converse directions and quasi-monomial equivalence, retaining only (i) => (iii)

extremal quasi-monomial valuation existence replaced by a global valuation-formula for every ideal

equality condition weakened to one-sided inequality

The stem does not explicitly reveal the correct option, and the definitions alone do not single out choice A. There is no obvious lexical or structural cue that leaks the answer.

The question is essentially a theorem-recall item: it asks which statement is equivalent to the valuation existence condition, and the correct option is the exact asymptotic equality appearing as the theorem’s companion condition. This makes it close to a direct restatement rather than an independently motivated problem.

There is some reasoning pressure because the distractors are nearby variants: a stronger universalization, a weaker inequality, a coordinatewise-inequality confusion, and replacement of asymptotic order by raw order. However, success mainly depends on knowing the precise theorem statement rather than generating a new argument.

The distractors are mathematically plausible and reflect realistic failure modes: strengthening to all partial products, weakening equality to inequality, confusing exact interpolation with lower bounds, and replacing the asymptotic invariant with the non-asymptotic one.

A technically strong but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses high-quality distractors, but it is largely tautological and only moderately tests generative reasoning.


Question:
Let \(\mathfrak a_1,\ldots,\mathfrak a_r\) be nonzero ideals of a ring \(R\), and let \(b_1,\ldots,b_r\in \mathbb R_{\ge 0}\). For \(\lambda\in \mathbb R_{\ge 0}\cup\{\infty\}\) and an ideal \(\mathfrak a\), define the filtration \((\lambda\cdot \mathfrak a)_\bullet=((\lambda\cdot \mathfrak a)_m)_{m\in\mathbb N}\) by
\[
(\lambda\cdot \mathfrak a)_m=
\begin{cases}
R,& \lambda=0,\\
\mathfrak a^{\lceil \lambda m\rceil},& 0<\lambda<\infty,\\
0,& \lambda=\infty.
\end{cases}
\]
If \((\mathfrak b_1)_\bullet,\ldots,(\mathfrak b_s)_\bullet\) are filtrations, their sum is the filtration \(\mathfrak c_\bullet\) with
\[
\mathfrak c_m=\sum_{m_1+\cdots+m_s=m}(\mathfrak b_1)_{m_1}\cdots(\mathfrak b_s)_{m_s}.
\]
Set
\[
I_\bullet:=\Big(\frac1{b_1}\cdot \mathfrak a_1\Big)_\bullet+\cdots+\Big(\frac1{b_r}\cdot \mathfrak a_r\Big)_\bullet,
\]
where \(1/0\) is interpreted as \(\infty\). For any ideal \(\mathfrak a\subseteq R\), define the order function
\[
\nu_I(\mathfrak a):=\sup\{m\in\mathbb N\mid \mathfrak a\subseteq I_m\},
\]
and the associated asymptotic Samuel function
\[
\bar\nu_I(\mathfrak a):=\lim_{k\to\infty}\frac{\nu_I(\mathfrak a^k)}{k}.
\]
Which statement holds for these data?

Choices:
B. The following three statements are equivalent: (i) there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); (ii) there exists a quasi-monomial valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); and (iii) for every nonempty subset \(J\subseteq\{1,\dots,r\}\), one has \(\bar\nu_I\big(\prod_{j\in J}\mathfrak a_j\big)=\sum_{j\in J} b_j\).
C. If there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\), then \(\bar\nu_I(\mathfrak a_1\cdots \mathfrak a_r)=\sum_{j=1}^r b_j\).
D. The following three statements are equivalent: (i) there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); (ii) there exists a quasi-monomial valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); and (iii) for every ideal \(\mathfrak a\subseteq R\), one has \(\bar\nu_I(\mathfrak a)=\inf_{v}\frac{v(\mathfrak a)}{v(I_\bullet)}\), where the infimum runs over all valuations \(v\) on \(R\) with \(v(I_\bullet)>0\).
E. The following three statements are equivalent: (i) there exists a valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); (ii) there exists a quasi-monomial valuation \(v\) on \(R\) such that \(v(\mathfrak a_j)=b_j\) for \(j=1,\dots,r\); and (iii) the inequality \(\bar\nu_I(\mathfrak a_1\cdots \mathfrak a_r)\ge \sum_{j=1}^r b_j\) holds.

The stem provides only definitions and setup; it does not explicitly state the equivalence in the correct option or otherwise single out choice A. There are no obvious lexical cues that reveal the answer.

The item is still fairly close to theorem recognition: the correct option is essentially the precise theorem statement in this notation. However, the presence of several nearby variants means it is not a pure verbatim restatement.

Some reasoning is needed to distinguish exact equivalence from weaker, stronger, or misdirected alternatives (equality vs inequality, full equivalence vs one implication, single product vs all subsets). But the task mainly tests recognition of the exact theorem rather than substantial generative problem solving.

The distractors are mathematically plausible and target realistic failure modes: overstrengthening the conclusion, weakening to a one-way implication, replacing equality by inequality, or swapping in an overly broad valuation formula. They are distinct and nontrivial.

A solid theorem-discrimination MCQ with no answer leakage and strong distractors, but it leans more toward precise theorem recall than genuine generative reasoning.