Related papers: Hilbert's 13th problem in prime characteristic

The algebraic form of Hilbert's 13th Problem asks for the resolvent degree $\text{rd}(n)$ of the general polynomial $f(x) = x^n + a_1 x^{n-1} + \ldots + a_n$ of degree $n$, where $a_1, \ldots, a_n$ are independent variables. The resolvent…

We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this…

Artin solved Hilbert's 17th problem, proving that a real polynomial in $n$ variables that is positive semidefinite is a sum of squares of rational functions, and Pfister showed that only $2^n$ squares are needed. In this paper, we…

Let $D$ be a negative integer congruent to $0$ or $1\bmod{4}$ and $\mathcal{O}=\mathcal{O}_D$ be the corresponding order of $ K=\mathbb{Q}(\sqrt{D})$. The Hilbert class polynomial $H_D(x)$ is the minimal polynomial of the $j$-invariant $…

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to $d$, where $d$ is a positive integer. In addition, we prove the following result which…

Let RD(n) denote the minimum d for which there exists a formula for the roots of the general degree n polynomial using only algebraic functions of d or fewer variables. In 1927, Hilbert sketched how the 27 lines on a cubic surface could be…

We give two elementary proofs, at a level understandable by students with only pre-calculus knowledge of Algebra, of the well known fact that an irreducible irrational n-th root of a positive rational number cannot be solution of a…

The second part of the Hilbert's sixteenth problem consists in determining the upper bound $\mathcal{H}(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $n\geq2$, it is still unknown…

For $S\subseteq \mathbb{F}^n$, consider the linear space of restrictions of degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted $\mathrm{h}_S(d,\mathbb{F})$, is the dimension of this space. We obtain a tight lower bound on…

Let $k({\bf x})=k(x_1,\ldots ,x_n)$ be the rational function field, and $k\subsetneqq L\subsetneqq k({\bf x})$ an intermediate field. Then, Hilbert's fourteenth problem asks whether the $k$-algebra $A:=L\cap k[x_1,\ldots ,x_n]$ is finitely…

Given an order, a commutative ring whose additive group is free of finite rank, a natural computational question is whether a fixed univariate polynomial $f \in \mathbb{Z}[X]$ has a root in this ring. In this paper, we show that the…

The Ehrhart polynomial of the $d$-th hypersimplex $\Delta(d,n)$ of order $n$ is studied. By computational experiments and a known result for $d=2$, we conjecture that the real part of every roots of the Ehrhart polynomial of $\Delta(d,n)$…

We consider the minimal free resolution of a generic set of n+1 forms (not necessarily of the same degree) in a polynomial ring of n variables. The Hilbert function for such an ideal is known, thanks to a result of Stanley and of Watanabe.…

I introduce a geometric interpretation of the set of standard graded Artinian Gorenstein algebras of codimension n and degree d: the standard locus, which is a subset of the projective space of degree d polynomials in n variables, and I…

Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been…

For each $n$, let $\text{RD}(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In 1945, Segre called for a better understanding of the large…

Let $d\ge4$ and $c\in(-d,d)$ be relatively prime integers. We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\le d\le 36$), the smallest prime $p\equiv c\pmod d$ with $p\ge(2dn-c)/(d-1)$ is the…

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+\infty otherwise. Let A_1,...,A_n be finite nonempty subsets of F, and let $$f(x_1,...,x_n)=a_1x_1^k+...+a_nx_n^k+g(x_1,...,x_n)\in…

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…

Let K be an algebraic function field of characteristic 2 with constant field C_K. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u,x of K with u…