A note on a paper by Hashemi and Kapur
Anna Nymann Heisel, Niels Lauritzen
TL;DR
The paper identifies a concrete counterexample to Hashemi and Kapur's truncated-polynomial Groebner basis conversion algorithm over a finite field, demonstrating that the method can fail to produce a complete $\prec_2$-Groebner basis. By constructing the setup with $R = \mathbb{F}_2[x,y,z]$, specific polynomials, and the truncation $tp_{1,2}$, the authors show that successive lifting via Gröbner coefficients can yield an incorrect, incomplete basis. A key finding is that the sequence in which S-polynomials are processed can affect whether a missing polynomial (e.g., $y^4$) appears in the final basis, revealing a flaw in the HK approach as implemented. The paper further notes that a more complex counterexample with additional variables exists, indicating the issue persists beyond the simplest case and undermines the reliability of the extended Buchberger algorithm in HK under standard division rules.
Abstract
Recently Hashemi and Kapur published an algorithm [1] for Groebner basis conversion by truncating polynomials according to a source and a target monomial order. Here we present a counterexample to this algorithm.