Selecting better separating linear forms for RUR efficiency

Develop methods to choose, for a given zero-dimensional ideal I ⊆ K[X1,…,Xn], a separating linear form t = Σi ti Xi (i.e., a linear form that is injective on V(I)) that optimizes the computation time, the coefficient size, and the practical usability of the reduced Rational Univariate Representation produced by the lexicographic Gröbner-basis-based algorithm described in the paper.

Background

The paper presents a Las Vegas algorithm that computes a reduced Rational Univariate Representation (RUR) from multiplication matrices (or Gröbner bases) by first selecting a separating linear form and then constructing bivariate lexicographic Gröbner bases to read off coordinate parametrizations. The choice of the separating form substantially affects sparsity, arithmetic cost, and the size of the resulting coefficients, particularly over Q, and thus also impacts downstream tasks such as numerical approximation and real root isolation.

Empirical results indicate that certified strategies can yield sparser separating forms and smaller RUR coefficients than random choices, but effects vary across examples. The authors emphasize that finding strategies that consistently optimize performance and output size is unresolved and explicitly identify it as an open problem.