Effective Bertini theorem and formulas for multiplicity and the local {\L}ojasiewicz exponent

TL;DR

This paper provides an effective version of Bertini's theorem, identifying specific hyperplanes with desired intersection properties, and derives formulas for multiplicity and the Łojasiewicz exponent of polynomial mappings.

Contribution

It introduces a method to explicitly find hyperplanes satisfying Bertini's theorem and offers effective formulas for multiplicity and Łojasiewicz exponent calculations.

Findings

Identifies a finite family of hyperplanes satisfying Bertini's conditions

Provides effective formulas for multiplicity of polynomial mappings

Derives formulas for the Łojasiewicz exponent

Abstract

The classical Bertini theorem on generic intersection of an algebraic set with hyperplanes states the following: \emph{Let X be a nonsingular closed subvariety of $\mathbb{P}^n_k$, where $k$ is an algebraically closed field. Then there exists a hyperplane $H\subset \mathbb{P}^n_k$ not containing $X$ and such that the scheme $H\cap X$ is regular at every point. Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system $|H|$ considered as a projective space. } We will show that one can effectively indicate a finite family of hyperplanes $H$ such that at least one of them satisfies the assertion of the Bertini theorem. As an application of the method used in the proof we will give effective formulas for the multiplicity and the {\L}ojasiewicz exponent of polynomial mappings.