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Thiago da Silva

Publications and source records attributed to Thiago da Silva.

10 recordsLinked to original sources
arXiv2026

LipschitzSaturation: A Macaulay2 Package for Computing Lipschitz Saturations of Modules and Toric Varieties

We introduce \verb|LipschitzSaturation|, a package for the computer algebra system \textit{Macaulay2} that implements algorithms for computing Lipschitz saturations of modules and toric varieties. In the module setting, the package handles three distinct saturation notions for an $\mathcal{O}_{X}$-submodule $\mathcal{M}\subseteq\mathcal{O}_{X}^{p}$: the 1-, 2-, and 3-Lipschitz saturations $\mathcal{M}_{S_{1}}$, $\mathcal{M}_{S_{2}}$ and $\mathcal{M}_{S_{3}}$, together with the auxiliary construction of the double module $\mathcal{M}_{D}$. To bypass the computationally intractable multivariate calculations for $\mathcal{M}_{S_{1}}$, we implemented a curve-based membership test, achieving near-constant runtime on parametric families that cause the purely algebraic method to time out or exhaust memory. For Toric Singularities, we implemented a construction algorithm. The package is freely available and requires \textit{Macaulay2} version 1.22 or later.

math.AG↗
arXiv2026

Bi-Lipschitz Invariants in Singularity Theory: Lojasiewicz Exponent and Euler Obstruction

In this work, we investigate the bi-Lipschitz invariance of two fundamental local invariants in singularity theory: the Łojasiewicz exponent and the local Euler obstruction. We draw inspiration from Bivià-Ausina and Fukui, whose framework we extend to ideals in rings of analytic functions defined on affine toric varieties. We establish conditions under which these invariants remain unchanged under bi-Lipschitz equivalence. We also provide an answer, to a particular case, to the open question of whether the local Euler obstruction is a bi-Lipschitz invariant. For hypersurfaces with isolated singularities, we show that the Euler obstruction is preserved under non-degeneracy conditions. These results contribute to the understanding of metric invariants in complex analytic geometry.

math.AG↗
arXiv2025

The projective analytic spectrum of the double of a module

In this work, we investigate the projectivized analytic spectrum of the double of a module, establishing some general properties, and we apply these results to $\mbox{Projan}(\cR((JM(X))_D))$ over the origin in $C\times C$, where $C$ is an irreducible curve in a hypersurface $X$.

math.AG↗
arXiv2025

Newton polyhedra and the integral closure of ideals on toric varieties

In this work, we extend Saia's results on the characterization of Newton non-degenerate ideals to the context of ideals in $O_{X(S)}$, where $X(S)$ is an affine toric variety defined by the semigroup $S\subset \mathbb{Z}^{n}_{+}$. We explore the relationship between the integral closure of ideals and the Newton polyhedron. We introduce and characterize non-degenerate ideals, showing that their integral closure is generated by specific monomials related to the Newton polyhedron.

math.AG↗
arXiv2023

Some remarks about $ρ$-regularity for real analytic maps

In this paper, we discuss the concept of $ρ$-regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map G with respect to the levels of a function $ρ$, which guarantees $ρ$-regularity. Consequently, both conditions are crucial for the presence of open book structures and the Milnor fibrations. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts.

math.DG↗