Let $I\subseteq\mathbb C[x_1,\ldots,x_n]$ be an ideal generated by homogeneous polynomials of a common degree $d$. We give a polynomial partial differential equation criterion guaranteeing that $-n/d$ is a root of the Bernstein-Sato polynomial $b_I(s)$. We apply this criterion to the ideal of maximal minors of a generic $m\times n$ matrix and obtain the distinguished root $-n$; combined with local divisibility along determinantal strata, this allows us to obtain the strong monodromy conjecture in the maximal-minor case. Finally, we prove that the criterion is stable under enlarging the linear span of the generators, adjoining generators in disjoint variables, products satisfying the natural slope condition, and Thom-Sebastiani sums. These stability results provide new classes of homogeneous ideals and polynomials for which the distinguished Bernstein-Sato root can be detected. Keywords. Bernstein-Sato polynomial, monodromy conjecture.
Let $f_m(x_0,\ldots,x_{2m-2}) =\det(x_{i+j})_{0\leq i,j\leq m-1}$ be the determinant of the generic symmetric Hankel matrix of size $m$. We determine its global Bernstein--Sato polynomial. The proof combines the nearby cycles of the corresponding secant hypersurface with an explicit Archimedean zeta integral over the c...
Let $X$ be a smooth complex affine variety of dimension $n$, and let $F=(f_1,\ldots,f_r)$ be a tuple of nonzero regular functions on $X$ such that $f:=\prod_{i=1}^r f_i$ is not invertible. We study the zero loci of the Bernstein-Sato ideals $B_F^{\mathbf a}$ for nonnegative integral shifts $\mathbf a$. For a fixed log...
Every quadratic Hessian slice of a Lorentzian polynomial yields bounded monomial ratios among the normalized coefficients of the polynomial. We determine exactly for which pairs $(n,d)$ these quadratic-slice ratios generate the full bounded-ratio cone for every $M$-convex support $S\subseteq\Delta_n^d$. For $d\geq 2$,...
Di-Jia Chen, Bo-Wen Gan, Ivy Liu et al.· 1 citation
Given a system $\mathbf{f}=(f_1,\ldots,f_n)$ of $n$ homogeneous forms in $n$ variables of the same degree, Macaulay's resultant vanishes precisely when the polynomials have a common projective zero. Its order of vanishing measures the singularity of the resultant hypersurface at $\mathbf{f}$. In this paper, we study ho...
M. L. Dogan, E. Tsigaridas, Z. Zafeirakopoulos· 0 citations
Let $q=p^k$, and let $A_1,\ldots,A_{r-1}$ be invertible $n\times n$ matrices over ${\mathbb F}_q$. We prove that, if $k\ge r$, there is a vector $x$ for which \[ x,A_1x,\ldots,A_{r-1}x \] are all nowhere zero. For $r=2$ this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomi...
Let $m\geq 2$ and $d\geq 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$. We prove the rational noncommutative Hodge conjecture for the dg category $\mathrm{MF}^{\mathrm{gr}...
Xun Lin, Shi-Zhuo Zhang· 0 citations
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