Verification of Gröbner basis candidates. (English) Zbl 1434.13029

Noro, Masayuki; Yokoyama, Kazuhiro

Hong, Hoon (ed.) et al., Mathematical software – ICMS 2014. 4th international congress, Seoul, South Korea, August 5–9, 2014. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 8592, 419-424 (2014).

Summary: We propose a modular method for verifying the correctness of a Gröbner basis candidate. For an inhomogeneous ideal \(I\), we propose to check that a Gröbner basis candidate \(G\) is a subset of \(I\) by computing an exact generating relation for each \(g\) in \(G\) by the given generating set of \(I\) via a modular method. The whole procedure is implemented in Risa/Asir, which is an open source general computer algebra system. By applying this method we succeeded in verifying the correctness of a Gröbner basis candidate computed in [V. G. Romanovski et al., J. Phys. A, Math. Theor. 40, No. 22, 5905–5919 (2007; Zbl 1127.34020)]. In their paper the candidate was computed by a black-box software system and it has been necessary to verify the candidate for ensuring the mathematical correctness of the paper.
For the entire collection see [Zbl 1293.65003].

MSC:

13P10	Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30	Symbolic computation and algebraic computation
13-04	Software, source code, etc. for problems pertaining to commutative algebra

Keywords:

Gröbner basis; modular algorithm; verification

Citations:

Zbl 1127.34020

Software:

Risa/Asir