Foundations of transcendental methods in computational nonlinear algebra

Introduction

Polynomial equations and inequalities raise fundamental theoretical issues, many of which have been answered by algebraic geometry. As for applications, nonlinearity is also a formidable computational challenge.

Proposed Methods

Based on recent proof-of-concept works, I propose new foundational methods in computational nonlinear algebra, motivated by the need for reliability and applicability. The joint development of theoretical aspects, algorithms, and software implementations will turn these proof-of-concepts into breakthroughs.

Theory Development

Concretely, I will develop a theory of transcendental continuation for the numerical computation of a wide range of multiple integrals. This will be based on a striking combination of:

1. Algebraic geometry
2. Symbolic algorithms
3. Numerical ODE solvers

This approach would enable the computation of many integrals (e.g., volume of semialgebraic sets or periods of complex varieties) with rigorous error bounds and high precision, exceeding thousands of digits.

Algorithm Design

Building upon transcendental continuation, I propose to design algorithms to compute certain algebraic invariants of complex varieties related to algebraic cycles and Hodge classes. This would extend far beyond the current reach of symbolic methods. This surprising application is backed by a recent success in Picard group computation.

Applications

Applications include:

• Algebraic geometry, with the development of computational tools to experiment on concrete examples and build databases that document the largest possible range of behavior.
• Diophantine approximations
• Feynman integrals
• Optimization