Intervals, Syzygies, Numerical Gröbner Bases: A Mixed Study

• Published in: Computer Algebra in Scientific Computing pp. 64 - 76
• Main Authors: Bodrato, Marco, Zanoni, Alberto
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2006
• Series: Lecture Notes in Computer Science
• Subjects: Gröbner bases; numerical coefficients; syzygies
• ISBN: 9783540451822; 354045182X
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11870814_5

In Gröbner bases computation, as in other algorithms in commutative algebra, a general open question is how to guide the calculations coping with numerical coefficients and/or not exact input data. It often happens that, due to error accumulation and/or insufficient working precision, the obtained result is not one expects from a theoretical derivation. The resulting basis may have more or less polynomials, a different number of solution, roots with different multiplicity, another Hilbert function, and so on. Augmenting precision we may overcome algorithmic errors, but one does not know in advance how much this precision should be, and a trial–and–error approach is often the only way to follow. Coping with initial errors is an even more difficult task. In this experimental work we propose the combined use of syzygies and interval arithmetic to decide what to do at each critical point of the algorithm. AMS Subject Classification: 13P10, 65H10, 90C31.