Deterministic methods identify key invariants in polynomial ideals
Deterministic Determination of Axial Constants, Sectional Regularities and Annihilator Numbers
Symbolic Computation
Summary
The paper deals with special mathematical properties of polynomial equations that change depending on how you look at them. The authors found clear ways to recognize the best conditions under which these properties behave consistently. They created step-by-step procedures using linear algebra—basically systematic calculations—to find these properties without relying on randomness. This helps ensure that the results hold deterministically and can be precisely computed.
What this means in practice
- •For computer algebra system developers: Implement deterministic routines to compute polynomial ideal invariants reliably without random coordinate choices.
- •For mathematics software engineers: Design better computational tools that detect stable positions in polynomial computations for more predictable algebraic geometry outputs.
Authors
Amir Hashemi, Rozhin Sadr, Werner M. Seiler
Abstract
Axial constants and sectional regularities are recently introduced invariants of homogeneous polynomial ideals that, like the already earlier introduced annihilator numbers, depend on generic choices of coordinates. We provide effective characterisations of the genericity underlying these invariants in terms of $β$-maximal position and strong stability. In particular, we show that $β$-maximal position suffices for determining generic annihilator numbers and sectional regularities, while its combination with strong stability yields corresponding results for axial constants. We also develop deterministic linear algebra procedures for detecting the required generic positions and for computing these invariants.