Graded Betti numbers of monomial ideals and code invariants
After having reviewed the basics of coding theory, I will introduce a family of invariants of an error correcting code called generalized weights. I will then show how to associate to a code a monomial ideal, which captures some of its properties and invariants. In particular, I will discuss how the graded Betti numbers of this ideal determine the generalized weights of the code. The talk is based on a joint work with Alberto Ravagnani.
Connections between Discrete Morse Theories
Morse theory is a powerful topological tool for efficiently analyzing a manifold by studying the behavior of a smooth scalar function defined on it. Thanks to their effective applications, two discretized versions of Morse theory gained a prominent role in the literature: piecewise-linear (PL) Morse theory [Banchoff 1967] and discrete Morse theory [Forman 1998]. In spite of their common origin in smooth Morse theory, the two approaches present several differences (from the domain and the definition of Morse function to the notion of critical point). Starting with the critical sets, this talk aims to investigate under which hypotheses it is possible to establish correspondences between these two theories and discuss which future developments can benefit from this relationship.
Multivariate cryptography and the complexity of solving a random polynomial system
A multivariate cryptograpic instance in practice is a multivariate polynomial system. So the security of a protocol rely on the complexity of solving a multivariate polynomial system. During this talk I will explain the invariant to which this complexity depends on: the solving degree. Unfortunately this invariant is hard to compute. We will talk about another invariant, the degree of regularity, that under certain condition, give us an upper bound on the solving degree. Finally we will talk about random polynomial systems and in particular what "random" means to us. We will give an upper bound on both the degree of regularity and the solving degree of such random systems.
30 years of collaboration with M.E. Rossi
We review the results we have found in collaboration with M.E. Rossi over the last 30 years and something more.
On the regularity of apolar 0-dimensional schemes and additive decompositions of polynomials
A 0-dimensional scheme is apolar to a homogeneous polynomial if its defining ideal is contained in the annihilator ideal of the given polynomial by acting as partial derivatives. In this talk, I will address the widely open problem investigating algebraic properties of minimal apolar schemes such as Hilbert function and regularity. I will present recent results from two joint works with E. Angelini and L. Chiantini, and with A. Bernardi and D. Taufer. In the first one, we explicitly construct, by means of liaison theory and Cayley-Bacharach properties, examples of polynomials admitting apolar sets of points of minimal cardinality with different regularity. In the second one, we address the open question of understanding whether all apolar 0-dimensional schemes of minimal length of degree-d polynomials are indeed regular in degree d. These questions are strictly related to questions regarding particular additive decompositions of polynomials.
Weakly Gorenstein ideals and the defining equations of Rees algebras
This talk is concerned with a classical problem in elimination theory, the determination of the implicit equations defining the graphs and images of rational maps between projective varieties. The problem amounts to identifying the torsion in the symmetric algebra of an ideal, and one technique to achieve this is based on a duality statement due to Jouanolou that expresses the torsion of a graded algebra in terms of a graded dual of this algebra. Unfortunately, Jouanolou duality requires the algebra to be Gorenstein, a rather restrictive hypothesis for symmetric algebras. In this talk, I will introduce a generalized notion of Gorensteinness, which we call weakly Gorenstein, and explain how Jouanolou duality can be extended to this larger class of algebras. The generalized duality leads to the solution of the implicitization problem for new classes of rational maps, and can be used in some cases to relate the implicitization problem for an ideal to the implicitization problem for one of its Fitting ideals. The talk is based on joint work with Yairon Cid-Ruiz, Bernd Ulrich, and Matthew Weaver.
A Wilf-type inequality for numerical semigroups and local rings
Numerical semigroups are additive submonoids of (N,+), with finite complement in N. Despite their simple structure, there are many interesting problems regarding numerical semigroups, often arising from their connection with other research subjects in mathematics.
In this talk, after giving a brief survey about the connections between numerical semigroups and one-dimensional local rings, I will present a longstanding conjecture known as Wilf’s conjecture, that is an inequality involving embedding dimension and multiplicity of a numerical semigroup. This conjecture has been proved for some classes of numerical semigroups, but no general strategy has been found to solve it. Successively, I will show a similar, but weaker, inequality, recently proved by A. Moscariello and myself.
This inequality, like also Wilf's conjecture, can be re-interpreted for one-dimensional local rings and also in higher dimension, provided that the conductor of the ring in its integral closure in the total ring of fractions is m-primary.
The new results presented in this talk are contained in two joint works with A. Moscariello and with C. A. Finocchiaro and A. Moscariello.
Limit F-signature Functions of Diagonal Hypersurfaces
We extend the theory of p-fractals introduced by Monsky and Teixeira to the setting of F-signature. We use this to prove that when p goes to infinity the F-signature function of diagonal hypersurfaces converges uniformily to a piecewise polynomial function. Moreover, we compute the F-signature of Fermat hypersurfaces. In particular, for the Fermat cubic in four variables we prove that the F-signature is strictly less than 1/8. This provides a negative answer to a question by Watanabe and Yoshida. This is a report on an ongoing project with S. Schideler, K. Tucker, and F. Zerman.