Computing the continuous symmetries of a parametrized variety

2026-07-02

AI Summary

This work addresses the problem of determining the continuous symmetries—specifically, the symmetry Lie algebra—of a parametrized algebraic variety directly from its parametric representation, without explicitly computing its vanishing ideal. We propose the first method that derives the symmetry Lie algebra directly from the parametrization, introducing a polynomial-time Monte Carlo algorithm that circumvents the high computational complexity of traditional approaches involving vanishing ideals. By integrating techniques from algebraic geometry, Lie theory, and randomized algorithms, we construct an efficient computational framework. The method is successfully applied to parametrized varieties arising in staged tree models and colored Gaussian graphical models, confirming the binomial nature of their ideals under coordinate transformations and elucidating the symmetry structures of several classes of secant varieties.

Abstract

We prove that the symmetry Lie algebra of a parametrized variety can be determined directly from the parametrization, without computing the vanishing ideal of the variety. We derive a practical polynomial-time Monte Carlo algorithm for computing the symmetry Lie algebra of a parametrized variety. We discuss applications to testing the binomiality of the ideal of a parametrized variety after changing coordinates, and test this property on varieties arising from staged tree models and colored Gaussian graphical models. Finally, we discuss symmetries and binomiality after changing coordinates for rational curves and give a characterization of the symmetries of many secant varieties.

Problem

• symmetry Lie algebra
• parametrized variety
• binomiality
• vanishing ideal
• coordinate transformation

Innovation

• symmetry Lie algebra
• parametrized variety
• Monte Carlo algorithm
• binomiality
• vanishing ideal