Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2402.15649

Some Lower Bounds on the Reach of an Algebraic Variety

Chris La Valle·Josué Tonelli-Cueto
View original source ↗Read source document ↗Export metadata ↓

Abstract

Separation bounds are a fundamental measure of the complexity of solving a zero-dimensional system as it measures how difficult it is to separate its zeroes. In the positive dimensional case, the notion of reach takes its place. In this paper, we provide bounds on the reach of a smooth algebraic variety in terms of several invariants of interest: the condition number, Smale's $γ$ and the bit-size. We also provide probabilistic bounds for random algebraic varieties under some general assumptions.

Explore related subjects

math.AGcs.CGcs.SC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chris La Valle, Josué Tonelli-Cueto. 2024-05-30. Some Lower Bounds on the Reach of an Algebraic Variety. https://arxiv.org/abs/2402.15649

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Nodal degeneration of chiral algebras I: Global structure and gluing formula

We define a natural extension of a universal factorization algebra $\mathcal{A}$ to families of stable punctured curves, by integrating over all semistable modifications. We prove that the resulting sheaf of factorization homology satisfies a natural gluing formula, by tensoring over a certain derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, generalizing the Verlinde formula for gluing of conformal blocks.

math.AG↗