arXiv subjects

Jesse Elliott

Publications and source records attributed to Jesse Elliott.

At least 19 recordsLinked to original sources

arXiv2026

On the equivalence of semidefinite programming and zero-sum semidefinite games

By results of Dantzig (1951) and Adler (2013), computing the optimal solutions of a linear program is equivalent to finding optimal strategies in zero-sum bimatrix games. Dantzig's original result was incomplete, in the sense that the reduction of a linear program to a zero-sum game did not work for all possible linear programs. We show that, under a natural constraint qualification requiring either the existence of strongly optimal primal-dual solutions or of a strictly unbounded direction, computing the solution of a semidefinite program is equivalent to finding optimal strategies in an associated zero-sum semidefinite game. Our work builds upon Ickstadt, Theobald, and Tsigaridas (2024), where, similar to Dantzig's work, the proposed reduction cannot handle a certain subclass of semidefinite programs. Our main proof ingredients for the equivalence result include: (i) a semidefinite generalization of von Stengel's (2023) extension of Dantzig's construction; (ii) techniques for handling more general duality phenomena in the semidefinite setting; and (iii) an explicit bound for the (coordinates) of the solutions of a semidefinite program. As a by-product, the game value provides a certificate: it is zero if and only if strongly optimal solutions exist, and otherwise optimal strategies yield an infeasibility certificate for the primal or dual program.

math.OC↗

arXiv2026

Primes of bad reduction for systems of polynomial equations

Consider polynomials $F_1,\dots,F_s$ in $\K[X_1,\dots,X_n]$ over a field $\K$, their zero-set $V(F_1,\dots,F_n)$ in $\Kbar^n$ and its decomposition into equidimensional components $V_0,\dots,V_n$ (with $V_i$ either empty or of dimension $i$ for all $i$). To each $V_i$, we can associate its Chow forms, which are polynomials in new variables $(U_{k,j})_{0\le k\le i, 0 \le j \le n}$, uniquely defined up to a scalar factor. These Chow forms completely characterize $V_i$: we can recover equations for $V_i$ from them, and their degree is $(i+1)$ times the degree of $V_i$. We discuss the situation when the $F_i$'s have integer coefficients, and study the question of when the Chow forms of the $V_i$'s defined as above can be reduced modulo $p$ to give Chow forms of the equidimensional components of $V(F_1 \bmod p,\dots,F_s \bmod p)$. We show that this is the case as soon as $p$ does not divide a certain nonzero integer $\Delta$ of height $O(n^{14} s h d^{3n+4})$, with $d$ and $h$ bounds on respectively the degrees and heights of the $F_i$'s.

math.AC↗

arXiv2025

Refined bit complexity for the computation of at least one point per connected component of a smooth complete intersection real algebraic set

We refine the bit complexity analysis of an algorithm for the computation of at least one point per connected component of a smooth real algebraic set, yielding exponential speedup (with respect to the number of variables) compared to prior works. The algorithm which is analyzed is based on the critical point method, reducing the problem to computations of critical points associated to the restriction of generic projections on lines to the studied variety. Our refinement, and the subsequent improved complexity statement, comes from a better utilization of the multi-affine structure of polynomial systems encoding these sets of critical points. The bit-size estimates on the size of the output produced by this algorithm are also improved by this refinement.

cs.SC↗

arXiv2025

Additive subgroups of a module that are saturated with respect to a subset of the ring

Let $T$ be a subset of a ring $A$, and let $M$ be an $A$-module. We study the additive subgroups $F$ of $M$ such that, for all $x \in M$, if $tx \in F$ for some $t \in T$, then $x \in F$. We call any such subset $F$ of $M$ a $T$-factroid of $M$, which is a kind of dual to the notion of a $T$-submodule of $M$. We connect the notion with the zero-divisors on $M$, various classes of primary and prime ideals of $A$, Euclidean domains, and the recent concepts of unit-additive commutative rings and of Egyptian fractions with respect to a multiplicative subset of a commutative ring. We also introduce a common generalization of local rings and unit-additive rings, called *sublocalizable* rings, and relate them to $T$-factroids.

math.RA↗

arXiv2024

Analytic Number Theory and Algebraic Asymptotic Analysis

This monograph elucidates and extends many theorems and conjectures in analytic number theory and algebraic asymptotic analysis via the natural notion of "degree" and a more general notion that we call "logexponential degree." Specifically, we define the \emph{degree} of a real function $f$ whose domain is not bounded above to be the infimum of all real numbers $t$ such that $f(x)$ is $O(x^t)$. The Riemann hypothesis, for example, is equivalent to the statement that the degree of the function $\pi(x)- \operatorname{li}(x)$ is $1/2$, where $\pi(x)$ is the prime counting function and $\operatorname{li}(x)$ is the logarithmic integral function; likewise, the abc conjecture is equivalent to the statement that a particular function has degree 1. Part 1 of the text is a survey of analytic number theory, Part 2 introduces the notion of logexponential degree and uses it to extend results in algebraic asymptotic analysis, and Part 3 applies the results of Part 2 to the various functions that figure most prominently in analytic number theory and Diophantine analysis. Central to the notion of logexponential degree are Hardy's \emph{logarithmico-exponential functions}, which are real functions defined in a neighborhood of $\infty$ that can be built from $\operatorname{id}$, $\exp$, and $\log$ using the operations $+$, $\cdot$, $/$, and $\circ$. Such functions are natural benchmarks for the orders of growth of functions in analytic number theory. The main goal of Part 3 is to express the logexponential degree of various functions in analytic number theory in terms of as few "logexponential primitives" as possible.

math.NT↗

arXiv2022

Bit complexity for computing one point in each connected component of a smooth real algebraic set

We analyze the bit complexity of an algorithm for the computation of at least one point in each connected component of a smooth real algebraic set. This work is a continuation of our analysis of the hypersurface case (On the bit complexity of finding points in connected components of a smooth real hypersurface, ISSAC'20). In this paper, we extend the analysis to more general cases. Let $F=(f_1,..., f_p)$ in $\mathbb{Z}[X_1, ... , X_n]^p$ be a sequence of polynomials with $V = V(F) \subset \mathbb{C}^n$ a smooth and equidimensional variety and $\langle F \rangle \subset \mathbb{C}[X_1, ..., X_n]$ a radical ideal. To compute at least one point in each connected component of $V \cap \mathbb{R}^n$, our starting point is an algorithm by Safey El Din and Schost (Polar varieties and computation of one point in each connected component of a smooth real algebraic set, ISSAC'03). This algorithm uses random changes of variables that are proven to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model; here, we analyze the bit complexity and the error probability, and we provide a quantitative analysis of the genericity statements. In particular, we are led to use Lagrange systems to describe polar varieties, as they make it simpler to rely on techniques such as weak transversality and an effective Nullstellensatz.

math.AG↗

arXiv2021

Asymptotic expansions of the prime counting function

We provide several asymptotic expansions of the prime counting function $π(x)$ and related functions. We define an {\it asymptotic continued fraction expansion} of a complex-valued function of a real or complex variable to be a possibly divergent continued fraction whose approximants provide an asymptotic expansion of the given function. We show that, for each positive integer $n$, two well-known continued fraction expansions of the exponential integral function $E_n(z)$ correspondingly yield two asymptotic continued fraction expansions of $π(x)/x$. We prove this by first establishing some general results about asymptotic continued fraction expansions. We show, for instance, that the "best"' rational function approximations of a function possessing an asymptotic Jacobi continued fraction expansion are precisely the approximants of the continued fraction, and as a corollary we determine all of the best rational function approximations of the function $π(e^x)/e^x$. Finally, we generalize our results on $π(x)$ to any arithmetic semigroup satisfying Axiom A, and thus to any number field.

math.NT↗

arXiv2021

Harmonic numbers and the prime counting function

We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these approximations are at least as good as the logarithmic integral approximation. As a corollary, we provide some reformulations of the Riemann hypothesis in terms of the prime counting function and the harmonic numbers.

math.NT↗

arXiv2020

Group actions, power mean orbit size, and musical scales

We provide an application of the theory of group actions to the study of musical scales. For any group $G$, finite $G$-set $S$, and real number $t$, we define the {\it $t$-power diameter} $\operatorname{diam}_t(G,S)$ to be the size of any maximal orbit of $S$ divided by the $t$-power mean orbit size of the elements of $S$. The symmetric group $S_{11}$ acts on the set of all tonic scales, where a {\it tonic scale} is a subset of $\mathbb{Z}_{12}$ containing $0$. We show that, for all $t \in [-1,1]$, among all the subgroups $G$ of $S_{11}$, the $t$-power diameter of the $G$-set of all heptatonic scales is largest for the subgroup $Γ$, and its conjugate subgroups, generated by $\{(1 \ 2),(3 \ 4),(5 \ 6),(8 \ 9),(10 \ 11)\}$. The unique maximal $Γ$-orbit consists of the 32 thāts of Hindustani classical music popularized by Bhatkhande. This analysis provides a reason why these 32 scales, among all 462 heptatonic scales, are of mathematical interest. We also apply our analysis, to a lesser degree, to hexatonic and pentatonic scales.

math.GR↗

arXiv2020

Asymptotic expansions of weighted prime power counting functions

We prove several asymptotic continued fraction expansions of $π(x)$, $Π(x)$, $\operatorname{li}(x)$, $\operatorname{Ri}(x)$, and related functions, where $π(x)$ is the prime counting function, $Π(x) = \sum_{k = 1}^\infty \frac{1}{k}π(\sqrt[k]{x})$ is the Riemann prime counting function, and $\operatorname{Ri}(x) = \sum_{k=1}^\infty \frac{ μ(k)}{k} \operatorname{li}(\sqrt[k]{x})$ is Riemann's approximation to the prime counting function. We also determine asymptotic continued fraction expansions of the function $\sum_{p \leq x} p^s$ for all $s \in \mathbb{C}$ with $\operatorname{Re}(s) > -1$, and of the functions $\sum_{a^x < p \leq a^{x+1}} \frac{1}{p}$ and $\log \prod_{a^x < p \leq a^{x+1}} (1 -1/p)^{-1}$ for all real numbers $a > 1$. We also determine the first few terms of an asymptotic continued fraction expansion of the function $π(ax)-π(bx)$ for $a > b > 0$. As a corollary of these results, we determine the best rational approximations of the "linearized" verions of these various functions.

math.NT↗

arXiv2016

Integer-valued polynomials on commutative rings and modules

The ring of integer-valued polynomials on an arbitrary integral domain is well-studied. In this paper we initiate and provide motivation for the study of integer-valued polynomials on commutative rings and modules. Several examples are computed, including the integer-valued polynomials over the ring $R[T_1,\ldots, T_n]/(T_1(T_1-r_1), \ldots, T_n(T_n-r_n))$ for any commutative ring $R$ and any elements $r_1, \ldots, r_n$ of $R$, as well as the integer-valued polynomials over the Nagata idealization $R(+)M$ of $M$ over $R$, where $M$ is an $R$-module such that every non-zerodivisor on $M$ is a non-zerodivisor of $R$.

math.AC↗

arXiv2015

Nuclei and applications to star, semistar, and semiprime operations

We show that the theory of quantales and quantic nuclei motivate new results on star operations, semistar operations, semiprime operations, ideal systems, and module systems, and conversely the latter theories motivate new results on quantales and quantic nuclei. Results include representation theorems for precoherent prequantales and multiplicative semilattices; characterizations of the simple prequantales; and a generalization to the setting of precoherent quantales of the construction of the largest finite type semistar operation and the largest stable semistar operation smaller than a given semistar operation.

math.RA↗

arXiv2015

Idempotent plethories

Let $k$ be a commutative ring with identity. A {\it $k$-plethory} is a commutative $k$-algebra $P$ together with a comonad structure $W_P$, called the {\it $P$-Witt ring} functor, on the covariant functor that it represents. We say that a $k$-plethory $P$ is {\it idempotent} if the command $W_P$ is idempotent, or equivalently if the map from the trivial $k$-plethory $k[e]$ to $P$ is a $k$-plethory epimorphism. We prove several results on idempotent plethories. We also study the $k$-plethories contained in $K[e]$, where $K$ is the total quotient ring of $k$, which are necessarily idempotent and contained in $\operatorname{Int}(k) = \{f \in K[e]: f(k) \subseteq k\}$. For example, for any ring $l$ between $k$ and $K$ we find necessary and sufficient conditions---all of which hold if $k$ is a integral domain of Krull type---so that the ring $\operatorname{Int}_l(k) = \operatorname{Int}(k) \cap l[e]$ has the structure, necessarily unique and idempotent, of a $k$-plethory with unit given by the inclusion $k[e] \longrightarrow \operatorname{Int}_l(k)$. Our results, when applied to the binomial plethory $\operatorname{Int}({\mathbb Z})$, specialize to known results on binomial rings.

math.AC↗

arXiv2014

Birings and plethories of integer-valued polynomials

Let $A$ and $B$ be commutative rings with identity. An {\it $A$-$B$-biring} is an $A$-algebra $S$ together with a lift of the functor $Hom_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An {\it $A$-plethory} is a monoid object in the monoidal category, equipped with the composition product, of $A$-$A$-birings. The polynomial ring $A[X]$ is an initial object in the category of such structures. The $D$-algebra $Int(D)$ has such a structure if $D = A$ is a domain such that the natural $D$-algebra homomorphism $θ_n: {\bigotimes_D}_{i = 1}^n Int(D) \longrightarrow Int(D^n)$ is an isomorphism for $n = 2$ and injective for $n \leq 4$. This holds in particular if $θ_n$ is an isomorphism for all $n$, which in turn holds, for example, if $D$ is a Krull domain or more generally a TV PVMD. In these cases we also examine properties of the functor $Hom_D(Int(D),-)$ from $D$-algebras to $D$-algebras, which we hope to show is a new object worthy of investigation in the theory of integer-valued polynomials.

math.AC↗

arXiv2012

Factoring formal power series over principal ideal domains

We provide an irreducibility test and factoring algorithm (with some qualifications) for formal power series in the unique factorization domain $R[[X]]$, where $R$ is any principal ideal domain. We also classify all integral domains arising as quotient rings of $R[[X]]$. Our main tool is a generalization of the $p$-adic Weierstrass preparation theorem to the context of complete filtered commutative rings.

math.AC↗

arXiv2011

Semistar operations on Dedekind domains

We give an explicit description of the lattice $\Semistar(D)$ of all semistar operations on any Dedekind domain $D$ from its set $\Max(D)$ of maximal ideals. This descpription is constructive if $\Max(D)$ is finite. As a corollary we show that $2^{n \choose [n/2]} \leq |\Semistar(D)| \leq 2^{2^n}$ if $n = |\Max(D)|$ is finite; we compute $|\Semistar(D)|$ if $|\Max(D)| \leq 7$; and we show that if $\Max(D)$ is infinite then $\Semistar(D)$ has cardinality $2^{2^{|\Max(D)|}}$.

math.AC↗

arXiv2011

Presentations and module bases of integer-valued polynomial rings

Let D be an integral domain with quotient field K. For any set X, the ring Int(D^X) of integer-valued polynomials on D^X is the set of all polynomials f in K[X] such that f(D^X) is a subset of D. Using the t-closure operation on fractional ideals, we find for any set X a D-algebra presentation of Int(D^X)$ by generators and relations for a large class of domains D, including any unique factorization domain D, and more generally any Krull domain D such that Int(D) has a regular basis, that is, a D-module basis consisting of exactly one polynomial of each degree. As a corollary we find for all such domains D an intrinsic characterization of the D-algebras that are isomorphic to a quotient of Int(D^X) for some set X. We also generalize the well-known result that a Krull domain D has a regular basis if and only if the Polya-Ostrowski group of D (that is, the subgroup of the class group of D generated by the images of the factorial ideals of D) is trivial, if and only if the product of the height one prime ideals of finite norm q is principal for every q.

math.AC↗

arXiv2011

Integer-valued polynomials, $t$-closure, and associated primes

Given an integral domain $D$ with quotient field $K$, the ring of integer-valued polynomials on D is the subring $\{f (X) \in K[X]: f(D) \subset D\}$ of the polynomial ring $K[X]$. Using the related tools of $t$-closure and associated primes, we generalize some known results on integer-valued polynomial rings over Krull domains, PVMD's, and Mori domains.

math.AC↗