We study the cone $\mathcal{M}_{n,2d}$ of nonnegative mean polynomials---real $n$-variate forms of degree $2d$ that can be expressed as weighted power means $M_{q,p}(Y,w)$ with $q>p$. This cone simultaneously generalises the cone of sums of squares $\Sigma_{n,2d}$ and the cone of sums of nonnegative circuit polynomials $\mathcal{C}_{n,2d}$. We prove that every square of an arbitrary polynomial belongs to the mean polynomial preprime $T_{\mathrm{mean}}$, that $T_{\mathrm{mean}}$ is strongly generating, and consequently that every polynomial strictly positive on a compact semialgebraic set admits a representation with mean polynomial certificates. We exhibit the Robinson form $\hat{R}$ as a separating example that lies in $\mathcal{M}_{4,4}$ but outside $\mathrm{SOSONC}_{4,4}$. Finally, we outline a convergent hierarchy of lower bounds for polynomial optimization based on the mean polynomial cone and discuss tractable depth-truncated approximations via signomial programming.
Let $X\subset{\mathbb P}^{n-1}$ be a hypersurface of degree $d\ge3$ with ordinary double points, where $n\ge3$. The roots of Bernstein-Sato polynomial of its defining polynomial $f$ are given up to sign by 1, $(n-1)/2$, and $j/d$ for $j\in{\mathbb Z}\cap[n,nd-n-p_f]$ with $p_f$ a positive integer. Here $p_f$ is bounded above by the minimal positive integer $q_s$ satisfying $\binom{q_s+n-1}{n-1}>s:=|{\rm Sing}\,X|$, and we can verify that $p_f$ coincides with $q_s$ in the case the singular points of $X$ are in ``general position". We show that this upper bound is sharp in the case $\binom{\lfloor d/2\rfloor+n-2}{n-1}\ge s$ or $\binom{d+n-3}{n-1}\ge sn$ by providing a homogeneous polynomial of degree $d$ such that the associated projective hypersurface has ordinary double points at given $s$ points in sufficiently general position and is nonsingular outside them (using a theorem of Alexander and Hirschowitz for the second case). It is conjectured that the above sharp bound under the first hypothesis can be extended naturally to the case where $X$ has only $A_2$-singularities instead of ordinary double points.
Given a field $\mathbb{F}$ and an integer $n \geq 2$, denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed, with the possible exception of finite fields with characteristic $2$ and less than $n-1$ elements. In this article, we prove that the equality $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ holds for all perfect non-quadratically closed fields with characteristic $2$ -- with the possible exception of fields with cardinality $2$ -- and for these fields we obtain a key result for a future analysis of the spaces that have the critical dimension $t_n(\mathbb{F})$.
We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified for every $n\geq 2$, with one remaining case: $n=2$, $q\equiv 1\pmod 4$, and $q$ not a square. We settle this case by proving that every positivity preserver on $M_2(\mathbb{F}_q)$ is injective on the set $\mathbb{F}_q^+$ of nonzero squares whenever $q\equiv 1\pmod 4$. The proof combines an idempotent reduction of positivity preservers with a well-known property of quadratic characters. This yields the complete classification of entrywise positivity preservers over every finite field and in every fixed dimension.
Dominique Guillot, Himanshu Gupta, P. K. Vishwakarma et al.· 0 citations
We study the simultaneous approximation of constant-degree polynomials over convex sets. For any family of $m$ degree-$d$ polynomials and any convex set ${H} \subseteq \mathbb{R}_{\ge0}^n$, we construct an $\epsilon$-Cover of the joint value set $\{(f_1(x), \dots, f_m(x)) : x \in {H}\}$ in the $\ell_\infty$-norm. This cover is of size $n^{O(\log(mn)/\epsilon^2)}$, provided the polynomials have constant range over the smallest $\ell_1$-ball inscribing ${H}$. Our approach extends classical net-based sparsifications for linear functions (e.g., Lipton, Markakis, and Mehta [2003]) to arbitrary families of constant-degree polynomials over general convex sets. We use a two-step scheme: first, we construct a quasi-polynomial pre-cover of the family on the smallest $\ell_1$-ball containing ${H}$ by using a concentration argument and leveraging a connection between Bernstein approximation and multinomial distributions; we then compress the pre-cover to ${H}$ by using a recursive degree reduction and feasibility programs anchored at points of the pre-cover. The existence of these covers immediately yields a unified framework for Quasi-Polynomial Time Approximation Schemes (QPTAS) across a wide range of a problems, including fixed-degree polynomial minimization over polyhedral sets, Constraint Satisfaction Problems (CSPs), Free Games, variational inequalities with polynomial operators (which implies guarantees for local Nash equilibria in polynomial games), and additive approximation for normalized densest $k$-subhypergraph on $O(1)$-uniform hypergraphs.
Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.· arXiv.org· 1 citation
Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form $\sum_{i=1}^{m} a_iX_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n \geq 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1, a_2, \ldots a_m \in \mathbb{Z}$, $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z})$. He gave bounds on $f(n)$ too. Koo and Lee further improved the bounds in 2025. This paper is organized as follows. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. In the first section we give necessary and sufficient condition for a quadratic form $\sum_{i=1}^{m}a_iX_i^2$ to be universal over $M_2(\mathbb{Z}_p)$ for $p=2$ and for an odd prime $p$. Consequently we express matrices of over $\mathbb{Z}_p$ as sum of squares. For a positive integer $n \geq 3$, we define $f(n)$ to be the smallest positive integer $m$ such that for every $p$-adic integers $a_1,a_2, \ldots a_m$ with at least three of them units, the diagonal quadratic form $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z}_p)$. In the next section we find bounds on $f(n)$ for $n \geq 3$.
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery--Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii--Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.
Egor D. Kosov, A. Zhukova· 1 citation· ⚡1
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