arXiv subjects

Gonzalo Manzano-Flores

Publications and source records attributed to Gonzalo Manzano-Flores.

5 recordsLinked to original sources

arXiv2026

Sums of squares in function fields over a henselian valued field

The Pythagoras number of a field is the smallest natural number $n$ such that every sum of squares in the field is a sum of $n$ squares. Given a nondyadic henselian valued field $(K, v)$, we show that the best bound on the Pythagoras number for function fields of curves over $K$ is essentially the same as for function fields of curves over the residue field $Kv$. As a consequence, we obtain the upper bound $5$ for the Pythagoras number of function fields of curves over any field $K$ for which the rational function field $K(X)$ has Pythagoras number $2$. The proof uses a reduction to the case where the value group is a subgroup of the real numbers, where then a local-global principle for quadratic forms due to V.~Mehmeti is applied.

math.NT↗

arXiv2026

The Automorphism groups of zero-dimensional monomial algebras

A monomial algebra B is defined as a quotient of a polynomial ring by a monomial ideal, which is an ideal generated by a finite set of monomials. In this paper, we determine the automorphism group of a monomial algebra B, under the assumption that B is a finite-dimensional vector space over a field of characteristic zero. We achieve this by providing an explicit classification of the homogeneous locally nilpotent derivations of B. The main body of the paper addresses the more general case of semigroup algebras, with the polynomial ring being a particular case.

math.AG↗

arXiv2026

The structure of automorphism groups of zero-dimensional monomial algebras

Let $A$ be a zero-dimensional monomial algebra over an algebraically closed field of characteristic zero, that is, a finite-dimensional quotient of a polynomial ring by a monomial ideal. Its automorphism group $G$ is a linear algebraic group, described through the homogeneous nilpotent derivations of $A$. We analyze the structure of $G$ in detail. Its identity component $G^0$ is a semidirect product of its unipotent radical and a reductive subgroup isomorphic to a product of general linear groups, and for each root degree we characterize when the associated derivations give rise to an additive root subgroup, and determine its dimension. Using the Lie brackets of these derivations, we then give an explicit algorithm that produces, out of the minimal monomial generators of the ideal, a family of root subgroups generating $G^0$ together with a maximal torus. Such a family is minimal in the generic case. We also show that the component group $G/G^0$ can be arbitrary: every finite group arises as the component group of the automorphism group of some zero-dimensional monomial algebra. Finally, we apply these results to the algebras $\mathbf{k}[\mathbf{x}]/\mathfrak{m}^d$, showing that the subgroup generated by a maximal torus and the outer root subgroups is exactly the subgroup of automorphisms with constant Jacobian determinant, and we deduce from this a new proof of Anick's theorem on the density of the tame automorphisms of $\mathbf{k}[\mathbf{x}]$.

math.AC↗

arXiv2026

Arithmetic genus inequalities with an application to sums of squares

We show variants of the genus inequality for the irreducible components of the special fiber of an arithmetic curve over a henselian discrete valuation ring of residue characteristic zero that take into account the non-existence of rational, respectively real points on the the components. We then apply this inequality to obtain the bound $2^{ng}$ (respectively $2^{n(g+1)}$) on the totally positive sum-of-two-squares index in the function field of a curve of genus $g$ over the field of $n$-fold iterated real Laurent series with (respectively without) real points. The bound $2^{n(g+1)}$ had been previously known only for hyperelliptic curves.

math.NT↗

arXiv2023

Sums of squares in function fields over henselian discretely valued fields

Let $n\in\mathbb{N}$ and let $K$ be a field with a henselian discrete valuation of rank $n$ with hereditarily euclidean residue field. Let $F/K$ be an algebraic function field in one variable. We show that the Pythagoras number of $F$ is $2$ or $3$ and we determine the order of the group of nonzero sums of squares modulo sums of two squares in $F$ in terms of the number of equivalence classes of discrete valuations on $F$ of rank at most $n.$ In the case of function fields of hyperelliptic curves of genus $g$, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by $2^{n(g+1)}$. We show in Example 4.6 that this bound is optimal.

math.NT↗