Dice Question Streamline Icon: https://streamlinehq.com

Beyond shape position: relations between lex Gröbner bases and multivariate subresultants

Ascertain whether there exist relations, beyond those already established for ideals in shape position, that systematically connect lexicographic Gröbner bases of zero-dimensional ideals with multivariate subresultant sequences; determine if the observed parallels indicate a broader general phenomenon.

Information Square Streamline Icon: https://streamlinehq.com

Background

The paper adapts subresultant-based parametrization strategies to the setting of lexicographic Gröbner bases in the bivariate case and exploits a specialization property (Gianni 1989). Motivated by apparent parallels between subresultants and lex Gröbner bases, the authors question whether these parallels signify a broader structural connection.

They report being unable to find general relations beyond known results that link ideals in shape position to multivariate subresultants (as in Cox 2023), leaving open whether a more general framework exists connecting these two paradigms.

References

One may wonder if the parallel between subresultant sequences and Gr"obner bases is a sign of a more general phenomenon. Unfortunately we couldn't find relations beyond what has been already found, relating ideals in shape position to multivariate subresultants in .

— Reading Rational Univariate Representations on lexicographic Groebner bases (2402.07141 - Demin et al., 11 Feb 2024) in Remark under “The bivariate case” in Section 2 (Reading parametrizations candidates on lexicographic Gröbner bases)