arXiv subjects

Francesca Gandini

Publications and source records attributed to Francesca Gandini.

7 recordsLinked to original sources

arXiv2026

An algorithm for invariants of elementary abelian groups

When we consider a finite abelian group acting linearly on a polynomial ring, we can find monomial generators for the subring of invariants. By Noether's degree bound and Hilbert's finiteness theorem, we know that there are finitely many minimal generators, but efficiently finding a generating set is not a trivial task. We present a new algorithm for computing the invariant ring for elementary abelian groups acting on polynomial rings with complex coefficients (or any other field of characteristic zero). We follow a two-step process: first we generate a collection of $n-k$ "seed" invariants by calculating the kernel of a weight matrix that encodes our action. After we find the seeds, we "grow" them into a generating set for the invariant ring by exploiting the lattice structure of invariants modulo $p$. Our algorithm performs better than the one currently available in Macaulay2, allowing us to compute invariants more quickly in this setting.

math.AC↗

arXiv2024

A Hilton-Milner theorem for exterior algebras

Recent work of Scott and Wilmer and of Woodroofe extends the Erd\H{o}s-Ko-Rado theorem from set systems to subspaces of k-forms in an exterior algebra. We prove an extension of the Hilton-Milner theorem to the exterior algebra setting, answering in a strong way a question asked by these authors.

math.CO↗

arXiv2023

Families of Harris Graphs

A Harris Graph is a tough, Eulerian, non-Hamiltonian graph. Several approaches to creating new Harris graphs from existing ones are explored, including creating families of Harris graphs and combining Harris graphs. Pictures of all Harris Graphs through order 9 and the number of Harris graphs through order 12 are included. We also prove a result about barnacle-free Harris graphs.

math.CO↗

arXiv2021

Toric and tropical Bertini theorems in positive characteristic

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the d-connectivity result for tropical varieties from that paper.

math.AG↗

arXiv2021

Degree bounds for invariant skew polynomials

When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal invariant polynomial is bounded above by the order of the group. Our work establishes that the same bound holds for invariant skew polynomials in the exterior algebra. Our approach to the problem relies on a theorem of Derksen that connects invariant theory to the study of ideals of subspace arrangements. We adapt his proof over the polynomial ring to the exterior algebra, reducing the question to establishing a bound on the Castelnuovo-Mumford regularity of intersections of linear ideals in the exterior algebra. We prove the required regularity bound using tools from representation theory. In particular, the proof relies on the existence of a functor on the category of polynomial functors that translates resolutions of ideals of subspace arrangements over the polynomial ring to resolutions of ideals of subspace arrangements over the exterior algebra.

math.AC↗

arXiv2020

The InvariantRing package for Macaulay2

We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2.

math.AC↗

arXiv2019

Resolutions of ideals associated to subspace arrangements

Given a collection of $t$ subspaces in an $n$-dimensional $\mathbb{K} $-vector space $W$ we can associate to them $t$ vanishing ideals in the symmetric algebra $\mathcal{S}(W^*) = \mathbb{K}[x_1,x_2,\dots,x_n]$. As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal. Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of $t$ linear ideals is equal to $t$. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of $t$ linear ideals is at most $t$ and they also showed that similar results hold for a more general class of ideals constructed from linear ideals. In this paper we show that analogous results hold when we replace the symmetric algebra $\mathcal{S}(W^*)$ with the exterior algebra $ \bigwedge(W^*)$ and work over a field of characteristic 0. To prove these results we rely on the functoriality of free resolutions and construct a functor $Ω$ from the category of polynomial functors to itself. The functor $Ω$ transforms resolutions of ideals in the symmetric algebra to resolutions of ideals in the exterior algebra.

math.AC↗