Seminars

Effective methods for multilinear birational transformations and contributions to polynomial data analysis

Pablo Gonzales-Mazon

Abstract:

This thesis explores two distinct subjects at the intersection of commutative algebra, algebraic geometry, multilinear algebra, and computer-aided geometric design:

1. The study and effective construction of multilinear birational maps
2. The extraction of information from measures and data using polynomials

The primary and most extensive part of this work is devoted to multilinear birational maps.
A multilinear birational map is a rational map $\phi: (\P^1)^n \dashrightarrow{} \P^n$, defined by multilinear polynomials, which admits an inverse rational map.
Birational transformations between projective spaces have been a central theme in algebraic geometry, tracing back to the seminal works of Cremona, which has witnessed significant advancement in the last decades.
Additionally, there has been a recent surge of interest in tensor-product birational maps, driven by the study of multiprojective spaces in commutative algebra and their practical application in computer-aided geometric design.

In the first part, we address algebraic and geometric aspects of multilinear birational maps.
We primarily focus on trilinear birational maps $\phi: (\P^1)^3 \dashrightarrow{} \P^3$, that we classify according to the algebraic structure of their space, base loci, and the minimal graded free resolutions of the ideal generated by the defining polynomials.
Furthermore, we develop the first methods for constructing and manipulating nonlinear birational maps in 3D with sufficient flexibility for geometric modeling and design.
Interestingly, we discover a characterization of birationality based on tensor rank, which yields effective constructions and opens the door to the application of tools from tensors to birationality.
We also extend our results to multilinear birational maps in arbitrary dimension, in the case that there is a multilinear inverse.

In the second part, our focus shifts to the application of polynomials in analyzing data and measures.
We tackle two distinct problems.
Firstly, we derive bounds for the size of $(1-\epsilon)$-nets for superlevel sets of real polynomials.
Our results allow us to extend the classical centerpoint theorem to polynomial inequalities of higher degree.
Secondly, we address the classification of real cylinders through five-point configurations where four points are cocyclic, i.e. they lie on a circumference.
This is an instance of the more general problems of real root classification of systems of real polynomials and the extraction of algebraic primitives from raw data.

Key words: birational map, multiprojective space, multilinear, syzygy, tensor, geometric modeling

Jury:

Reviewers:

• Alicia DICKENSTEIN, Professor emeritus, Universidad de Buenos Aires, Argentine
• Daniele FAENZI, Professor, Université de Bourgogne, France
• Hal SCHENCK, Professor, Auburn University, Alabama, USA

Examiners:

• Julie DÉSERTI, Professor, Université d’Orléans, France
• Andreas HÖRING, Professor, Université Côte d’Azur, France
• Sonia PÉREZ-DÍAZ, Professor, Universidad de Alcalá, Spain

Supervisor:

• Laurent BUSÉ, Research Director, Centre Inria d’Université Côte d’Azur, France