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 $t_n(\mathbb{F})$.
Let $n \geq 2$ be a positive integer, and $q$ be a prime power. We study the $number$ of additive decompositions of nonscalar matrices in the matrix ring $M_n(\mathbb{F}_q)$ as sums of elements from two prescribed conjugacy classes. Let $z \in M_n(\mathbb{F}_q)$ be nonscalar. We show that, except for the case $(n,q,\te...
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...
For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $\Omega \subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only...
We determine the complete ordered solution set of the Fermat-type matrix equation $X^n+Y^n=Z^n$ in $\mathrm{GL}_2(\mathbb{Z})$ for every integer $n\ge3$, expressed in terms of simultaneous integral-conjugacy orbits while keeping the variable symmetries of the equation separate. The equation is solvable if and only if $...
We resolve the Rademacher singularity conjecture: an $n\times n$ matrix with independent uniform $\{-1,1\}$ entries is singular with probability $(2+o(1))n^22^{-n}$. More generally, for an $n\times n$ matrix $M_n$ with independent entries uniform on a fixed set $S\subset\mathbb{R}$ of cardinality $q\ge2$, we prove $\ma...
Fix $I = (0,\rho)$, where $0<\rho\leq\infty$, and let $\mathbb{P}_n(I)$ be the set of positive semidefinite $n\times n$ matrices with entries in $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \to \mathbb{R}$ for which the entrywise calculus $f[A] = [f(a_{ij})]_{i,j = 1}^{n}$ preserv...
Ludovick Bouthat, Dominique Guillot· 0 citations
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