Universal denominators of Hilbert series

On the absence of uniform denominators in Hilbert's 17th problem

Hilbert showed that for most $(n,m)$ there exist psd forms $p(x_1,...,x_n)$ of degree $m$ which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form $h$ so that $h^2p$ is a sum of squares of forms; that is, $p$ is a sum of squares of rational functions with denominator $h$. We show that, for every such $(n,m)$ there does not exist a single form $h$ which serves in this way as a denominator for {\it every} psd $p(x_1,...,x_n)$ of degree $m$.

A new algorithm for the recursion of multisums with improved universal denominator

The purpose of the paper is to introduce two new algorithms. The first one computes a linear recursion for proper hypergeometric multisums, by treating one summation variable at a time, and provides rational certificates along the way. A key part in the search of a linear recursion is an improved universal denominator algorithm that constructs all rational solutions $x(n)$ of the equation $$ \frac{a_m(n)}{b_m(n)}x(n+m)+...+\frac{a_0(n)}{b_0(n)}x(n)= c(n),$$ where $a_i(n), b_i(n), c(n)$ are polynomials. Our algorithm improves Abramov's universal denominator.

Equivariant Hilbert Series of Monomial Orbits

The equivariant Hilbert series of an ideal generated by an orbit of a monomial under the action of the monoid $\mbox{Inc}(\mathbb{N})$ of strictly increasing functions is determined. This is used to find the dimension and degree of such an ideal. The result also suggests that the description of the denominator of an equivariant Hilbert series of an arbitrary $\mbox{Inc}(\mathbb{N})$-invariant ideal as given by Nagel and Römer is rather efficient.

G-集合范畴表示之希尔伯特级数

Hilbert series of representations of categories of $G$-sets

本文研究有限群 $G$ 的有限自由 $G$-集合范畴上反变函子的希尔伯特级数，证明其具有有理形式，且分母线性多项式的系数位于由 $G$ 的特征标表生成的域中。

Rationality of Equivariant Hilbert Series and Asymptotic Properties

An FI- or an OI-module $\mathbf{M}$ over a corresponding noetherian polynomial algebra $\mathbf{P}$ may be thought of as a sequence of compatible modules $\mathbf{M}_n$ over a polynomial ring $\mathbf{P}_n$ whose number of variables depends linearly on $n$. In order to study invariants of the modules $\mathbf{M}_n$ in dependence of $n$, an equivariant Hilbert series is introduced if $\mathbf{M}$ is graded. If $\mathbf{M}$ is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the ir…

Universal Lex Ideal Approximations of Extended Hilbert Functions and Hamilton Numbers

Let $R^h$ denote the polynomial ring in variables $x_1,\,\ldots,\, x_h$ over a specified field $K$. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with $x_1 > \cdots > x_h$. Given a fixed homogeneous ideal $I$ in $R^h$, for each $d$ there is unique lex ideal generated in degree at most $d$ whose Hilbert function agrees with the Hilbert function of $I$ up to degree $d$. When we consider $IR^N$ for $N \geq h$, the set $\mathfrak{B}_d(I,N)$ of minimal generators for this lex ideal in degree at most $d$ may change, but $\mathfrak{B}_d(I,N)$ is con…