Determining the properties of a random polynomial has fuelled significant investigation over the past century. One driving force of this research is a 1936 paper of Van der Waerden. Fix $n \geq 3$ and let $E_n(B)$ be the number of monic, irreducible, non-$S_n$ polynomials $f = X^n + a_1 X^{n-1} + \cdots + a_n$ with $|a_j| \leq B$ for all $j$. A recent breakthrough of Bhargava bounds $E_n(B) \ll B^{n-1}$. This spectacularly resolves a conjecture of Van der Waerden, but leaves open its stronger form, namely that $E_n(B) = o(B^{n-1})$. Inspired by recent progress, we now address this strong form. Bhargava's result, together with work of Chow and Dietmann, essentially reduces the strong Van der Waerden conjecture to the claim that the number of polynomials $f$ with Galois group $A_n$ is $o(B^{n-1})$. Assuming a twisted function field version of Chowla's conjecture, we prove this claim. This not only connects two active and challenging areas of research, but also conditionally resolves the strong Van der Waerden conjecture for all $n \geq 7$. Our proof is based on a variant of Heath-Brown and Pierce's square sieve and $q$-van der Corput differencing. Our methods also apply to the analogous problem of counting square discriminants of polynomials that are not necessarily monic. In addition to describing our new contributions, we briefly elaborate on the various conjectures appearing in Van der Waerden's paper and some of the exciting recent work of others in this area of arithmetic statistics.
We show that the weak Markus--Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on $\R^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on $\R^{18}$. The dimensions $3\leq n\leq13$ remain open.
Álvaro Castañeda, Gerardo Honorato, Francisco Valenzuela-Henríquez· 0 citations
Let $G=(V,E)$ be a simple graph of order $n$ and let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ be the eigenvalues of its Laplacian matrix. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le |E|+\binom{k+1}{2}$, which was recently confirmed by Kothari and Tudose. Before Brouwer's conjecture was proved, Lew (JCT-B, 2026) established a weaker form of Brouwer's Laplacian eigenvalue inequality and proposed two conjectures for upper bounds on the sum of the $k$ largest Laplacian eigenvalues, one in terms of the matching number and the other in terms of the vertex-cover number. Using Brouwer's Laplacian inequality, we prove both conjectures.
Let $X$ be a smooth complex projective variety of dimension $n$, and let $L$ be an ample Cartier divisor. We prove that $K_X+mL$ is globally generated for every integer $m\geq\lceil C_0n\rceil$, where $C_0=1.77629\ldots$ is an explicit constant. In particular, $K_X+2nL$ is globally generated. Our main input is a new estimate for the multiplicity of a minimal log canonical center. If $(X,\Delta)$ is log canonical near a closed point $x$ but is not klt at $x$, and $W$ is the positive-dimensional minimal log canonical center through $x$, then $2\overline{e}_1(\mathfrak m_{W,x})\leq\bigl(\dim W-\operatorname{lct}_x((X,\Delta);\mathfrak m_x)\bigr)\operatorname{mult}_xW$, where $\overline{e}_1(\mathfrak m_{W,x})$ is the first normal Hilbert coefficient of the maximal ideal of $\mathcal O_{W,x}$. This implies $\operatorname{mult}_xW\leq \frac{(a+c)^{a+c}}{a^ac^c}$, where $a:=\frac{\dim W-\operatorname{lct}_x((X,\Delta);\mathfrak m_{x})}{2}$ and $c:=\operatorname{edim}\mathcal O_{W,x}-\dim W$.
The Casas--Alvero conjecture asserts that a monic polynomial $f$ of degree $d$ over a field of characteristic zero sharing a non-constant factor with each of $f',\dots,f^{(d-1)}$ is the $d$-th power of a linear polynomial. Two reductions are carried out. Once a root is translated to the origin, the condition at index $i$ is vacuous unless the coefficient $a_{d-i}$ is nonzero, so the Casas--Alvero locus stratifies by the support of the centred normal form; prescribing in addition which root realises each surviving condition makes the system unit lower triangular and eliminates the coefficients of $f$ entirely. When one root suffices, what remains is a single integer, and that integer is MacMahon's determinant: the number of permutations of $\{1,\dots,d\}$ with descent set the support of $f$. Since every subset is a descent set, no counterexample has fewer than three recycled roots; in characteristic $p$ a prime dividing one of these counts but none of the accompanying integers $z_{i}$ is a bad prime, the largest count in degree $d$ being the Euler zigzag number $A_{d}$, so that the irregular prime $691$ is bad for $d=11$. The stratification also yields short proofs of the two characteristic-$p$ propositions on which all known cases rest; supports of two elements are settled separately in every characteristic, whence a counterexample has at least four terms in centred normal form and a further Gr\"obner-free criterion for bad primes.
One of the authors (Chen) had previously proved a result on the behavior of Betti-Whittaker periods under duality for cohomological cuspidal automorphic representations of ${\rm GL}_n/{\mathbb Q}$ under some regularity assumptions while using their relation to $L$-values as an anchor in his proof. In this article we prove a generalization of this result to ${\rm GL}_n$ over any number field $F$ without any regularity assumptions and without recourse to $L$-values, while using the outer-automorphism of ${\rm GL}_n$ as the main tool. Then, using results of Harder and one of the other authors (Raghuram), we give applications to new rationality results for the ratios of special values of general triple product $L$-functions and for general twisted Asai $L$-functions. We also give a new proof of a previous result of Bhagwat and Raghuram on the special values of $L$-functions for orthogonal groups. We present variations on period relations for the Betti-Shalika periods under duality, and the behavior of Betti-Whittaker periods under Galois automorphisms of $F$.
Giancarlo Castellano, Shih-Yu Chen, N. Darshan et al.· 1 citation
For $n>2$ and $k>1$, define the polynomial \[F_{n,k}(x) = \Phi_n(1 + x + \cdots + x^k),\] where $\Phi_n$ denotes the $n$-th cyclotomic polynomial. The \emph{cyclotomic conjecture} proposed by Gimbert (1999) exactly describes the irreducibility of $F_{n,k}(x)$ over $\mathbb{Q}$ in terms of $n$ and $k$. Conde, Gimbert, Gonz\'{a}lez, Miller and Miret (2014) established that the cyclotomic conjecture, if true, would imply the non-existence of almost Moore digraphs - a well-known open question concerning the directed degree-diameter problem. In this article, we prove the cyclotomic conjecture and, as a consequence, show that there are no almost Moore digraphs with maximum out-degree $d$ and diameter $k$ for any $d>1$ and $k>2$.
