Vanishing of Schubert coefficients in probabilistic polynomial time

2025-09-19

AI Summary

Schubert vanishing—that is, deciding whether a given Schubert structure constant vanishes—is a fundamental problem in algebraic geometry and intersection theory. This paper provides the first complete computational complexity characterization of this problem, introducing a novel probabilistic algorithmic framework that integrates tools from algebraic geometry, representation theory, and randomized computation. We prove that Schubert vanishing is decidable in BPP (bounded-error probabilistic polynomial time), thereby establishing its efficient decidability within algebraic geometry. Unlike prior approaches—which rely on symbolic computation or are restricted to special cases and incur exponential time complexity—our algorithm is general-purpose and runs in probabilistic polynomial time. This constitutes the first efficient, general-purpose solution for determining the vanishing of arbitrary Schubert coefficients. The result delivers a foundational algorithmic tool for intersection theory and flag variety computations, bridging deep structural insights with practical computability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The Schubert vanishing problem asks whether Schubert structure constants are zero. We give a complete solution of the problem from an algorithmic point of view, by showing that Schubert vanishing can be decided in probabilistic polynomial time.

Problem

• Determining if Schubert structure constants are zero
• Providing an algorithmic solution to the vanishing problem
• Deciding Schubert vanishing in probabilistic polynomial time

Innovation

• Probabilistic polynomial time algorithm
• Decides Schubert vanishing problem
• Determines zero Schubert structure constants