Related papers: Hilbert's 13th Problem for Algebraic Groups

The resolvent degree $\textrm{rd}_{\mathbb{C}}(n)$ is the smallest integer $d$ such that a root of the general polynomial $$f(x) = x^n + a_1 x^{n-1} + \ldots + a_n$$ can be expressed as a composition of algebraic functions in at most $d$…

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…

Resolvent degree ($\operatorname{RD}$) is an invariant of finite groups in terms of the complexity of their algebraic actions. We address the problem of bounding $\operatorname{RD}(G)$ for all finite simple groups using the methods…

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…

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…

Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group $G$ has only been investigated when $G$ is a…

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…

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…

For each $n$, let 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 this paper, we recover an algorithm of Sylvester for determining…

In 1683 Tschirnhaus claimed to have developed an algebraic method to determine the roots of any degree $n$ polynomial. His argument was flawed, but it spurred a great deal of work by mathematicians including Bring, Jerrard, Hamilton,…

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…

Hilbert's 14th problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the field of fractions of the algebra. It has a negative answer due to the counterexample of Nagata.…

Let $R={\sf k}[x,y,z]$, the polynomial ring over a field $\sf k$. Several of the authors previously classified nets of ternary conics and their specializations over an algebraically closed field. We here show that when $\sf k$ is…

This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with…

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f…

Hilbert's Nullstellensatz is a fundamental result in algebraic geometry that gives a necessary and sufficient condition for a finite collection of multivariate polynomials to have a common zero in an algebraically closed field. Associated…

We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…

For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$…

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…

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…