Bounding the degrees of a minimal \(\mu\)-basis for a rational surface parametrization. (English) Zbl 1470.13022
Summary: In this paper, we study how the degrees of the elements in a minimal \(\mu\)-basis of a parametrized surface behave. For an arbitrary rational surface parametrization \(P(s, t) = (a_1(s, t), a_2(s, t), a_3(s, t), a_4(s, t)) \in \mathbb{F} [s, t]^4\) over an infinite field \(\mathbb{F}\), we show the existence of a \(\mu\)-basis with polynomials bounded in degree by \(O(d^{33})\), where \(d = \max(\deg(a_1), \deg(a_2), \deg(a_3), \deg(a_4))\). Under additional assumptions we can obtain tighter bounds.
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 14Q10 | Computational aspects of algebraic surfaces |
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
syzygies; \(\mu\)-basis; Koszul complex; Quillen-Suslin theorem; unimodular matrix; liaisonSoftware:
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