Strong $\mu$-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

Abstract

Abstract. We investigate conditions under which the resultant of a µ-basis for a rational tensor product surface is the implicit equation of the surface without any extraneous factors. In this case, we also derive a formula for the implicit degree of the rational surface based only on the bidegree of the rational parametrization and the bidegrees of the elements of the µ-basis without any knowledge of the number or multiplicities of the base points, assuming only that all the base points are local complete intersections. We conclude that in this case the implicit degree of a rational surface of bidegree (m, n) is at most mn, so the rational surface must have at least mn base points counting multiplicity. When the resultant of a µ-basis generates extraneous factors, we show how to predict and compute these extraneous factors from either the existence of bad base points or anomalies occurring in the parametrization at infinity. Examples are provided to flesh out the theory.

Syzygies for translational surfaces

“…Furthermore, by Lemma 2.5, the columns of N f [P Q R] and N g [p q r] are three linearly independent syzygies. Hence the conclusion that none of the base points of h(s, u; t, v) are local complete intersections follows immediately byShen and Goldman (2017a), Lemma 2.1 or by, Lemma 3.2. ✷ Proposition 3.2.…”

“…It is often difficult to implicitize a surface that has a complicated collection of base points. In Section 3, we will use the algorithm in Shen and Goldman (2017a) to compute the implicit equation of a translational surface h from the resultant of a µ-basis for h.…”

“…Then the extraneous factor E(x) containsP (s 0 ,u 0 ;t 0 ,v 0 ) ) • x as an extraneous linear factor induced by the base point (s 0 , u 0 ; t 0 , v 0 ), as well as extraneous factors associated to infinity, that is, to points where uv = 0. Refer to Theorem 17 (Shen and Goldman, 2017a) to compute the extraneous factors associated to infinity.…”

“…If Res(P ′ • x,Q ′ • x,R ′ • x) ≡ 0, then by Corollary 3.3, rank[P ′ ,Q ′ ,R ′ ] (s 0 ,u 0 ;t 0 ,v 0 ) ) = 1 for every base point (s 0 , u 0 ; t 0 , v 0 ). Now the claim follows directly from Shen-Goldman (2017a). ✷ If Res(P ′ •x,Q ′ •x,R ′ •x) ≡ 0, we can no longer apply resultants with our special syzygies or µ-bases for translational surface to compute the implicit equation.…”

“…Geometrically, the case when the resultant is identically zero is related to the complexity of the base points. We refer readers to the papers by Goldman (2017a) and(2017b) for a detailed study of implicitization of tensor product surfaces with bad base points using the resultant of three moving planes.…”