Periodic Reporting for period 1 - 10000 DIGITS (Foundations of transcendental methods in computational nonlinear algebra)
Reporting period: 2022-04-01 to 2024-09-30
Polynomial equations and inequalities raises fundamental theoretical issues, many of which have been answered by algebraic geometry. As of applications, nonlinearity is also a formidable computational challenge. 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. Concretely, I will develop a theory of transcendental continuation for the numerical computation of a wide range of multipl integrals, based on a striking combination of algebraic geometry, symbolic algorithms and numerical ODE solvers. This 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, more than thousands of digits. Building upon transcendental continuation, I propose to design algorithms to compute certain algebraic invariants of complex varieties related to algebraic cycles and Hodge classes, far beyond the current reach of symbolic methods. This surprising application is backed by a recent success on Picard group computation. 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. Besides, I propose applications to Diophantine approximations, Feynman integrals, and optimization.
The team made significant progress on three main themes.
On polynomial systems we worked with Rafal Mohr (Sorbonne U), Mohab Safey El Din (Sorbonne U) and Christian Eder (U Kaiserslautern). We obtained, new efficient algorithms for equidimensional decomposition of polynomial ideals. This makes it possible to better handle polynomial systems from applications which often feature spurious components of higher dimension reflecting singular positions.
On symbolic integration, Hadrien Brochet started a PhD thesis (ERC-funded) and, as an early result, obtained a new summation algorithm (with Bruno Salvy, Inria). He develops software libraries in Maple and Julia that enables applications.
On the computation of periods of algebraic varieties, the thesis of Eric Pichon-Pharabod yields new computational methods based on an effective theory of Lefschetz fibrations (collaboration with Pierre Vanhove, CEA). They have been applied to some Feynman integrals.
On polynomial systems we worked with Rafal Mohr (Sorbonne U), Mohab Safey El Din (Sorbonne U) and Christian Eder (U Kaiserslautern). We obtained, new efficient algorithms for equidimensional decomposition of polynomial ideals. This makes it possible to better handle polynomial systems from applications which often feature spurious components of higher dimension reflecting singular positions.
On symbolic integration, Hadrien Brochet started a PhD thesis (ERC-funded) and, as an early result, obtained a new summation algorithm (with Bruno Salvy, Inria). He develops software libraries in Maple and Julia that enables applications.
On the computation of periods of algebraic varieties, the thesis of Eric Pichon-Pharabod yields new computational methods based on an effective theory of Lefschetz fibrations (collaboration with Pierre Vanhove, CEA). They have been applied to some Feynman integrals.
The highlight of the first reporting period is the new algorithm for computing periods of projective hypersurfaces obtained with Eric Pichon-Pharabod and Pierre Vanhove. It validates the general strategy and points at necessary progress to be made concerning the numerical high-precision integration of linear ODEs and effective approach to the topology of complex varieties.