Carlos D'Andrea

IMPLICITIZATION OF SURFACES IN ℙ3 IN THE PRESENCE OF BASE POINTS

We show that the method of moving quadrics for implicitizing surfaces in ℙ3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit equation can be computed as the resultant of the first syzygies.

Inversion of parameterized hypersurfaces by means of subresultants

We present a subresultant-based algorithm for deciding if the parametrization of a toric hypersurface is invertible or not, and for computing the inverse of the parametrization in the case where it exists. The algorithm takes into account the monomial structure of the input polynomials.

Hybrid Sparse Resultant Matrices for Bivariate Polynomials

We study systems of three bivariate polynomials whose Newton polygons are scaled copies of a single polygon. Our main contribution is to construct square resultant matrices, which are submatrices of those introduced by Cattaniet al. (1998), and whose determinants are nontrivial multiples of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict our attention to matrices of (almost) Sylvester-type and systems as specified above, then the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination, namely by a new piecewise-linear lifting. The major motivation comes from systems encountered in geometric modeling. Our preliminary Maple implementation, applied to certain examples, illustrates our construction and compares it with alternative matrices.

Macaulay style formulas for toric residues

We present an explicit formula for computing toric residues of ample divisors as a quotient of two determinants, à la Macaulay, where the numerator is a minor of the denominator. We present a combinatorial construction of a specific element of residue 1. We also give an irreducible representation of toric residues by extending the theory of subresultants to monomials of critical degree in the homogeneous coordinate ring of the corresponding toric variety.

Hybrid sparse resultant matrices for bivariate systems

Our main contribution is an explicit construction of square resultant matrices, which are submatrices of those introduced by Cattani, Dickenstein and Sturmfels [4]. The determinant is a nontrivial multiple of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict attention to such matrices, the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination. The algorithm uses a new piecewise-linear lifting, defined for bivariate systems of 3 polynomials with Newton polygons being scaled copies of a single polygon. The major motivation comes from systems encountered in CAD. Our MAPLE implementation, applied to certain examples, illustrates our construction and compares with alternative matrices.

The Broken Spaghetti Noodle

A popular question in recreational mathematics is the following: If we drop a spaghetti noodle and it breaks at two random places, what is the probability that we can form a triangle with the three resulting segments? See, for example, [2, chap. 1, sec. 6], [3, p. 6], [4, p. 31], or [7, pp. 30–36]. This is an elementary problem in geometric probability. Clearly the length of the noodle (or equivalently our choice of unit length) does not matter, so the problem amounts to choosing two numbers at random from the interval (0, 1), say a and b with a < b, and looking at the resulting intervals (0, a), (a, b), and (b, 1). We will be able to form a triangle when the positive numbers a, b − a, and 1 − b satisfy the triangle inequality (i.e., when no interval is longer than the combined lengths of the other two). Equivalently, this will be the case when all three intervals have length less than 1/2. Therefore, a triangle can be formed precisely when the following three inequalities hold: a < 1/2, b − a < 1/2, and b > 1/2. Figure 1 shows all possible outcomes 0 < a < b < 1, and the darker shaded region consists of all “favorable” outcomes, when a triangle can be formed. Comparing areas, we see that the probability of succeeding in getting a triangle is 1/4.

Subresultants and generic monomial bases

Given n polynomials in n variables of respective degrees d1,...,dn, and a set of monomials of cardinality d1...dn, we give an explicit subresultant-based polynomial expression in the coefficients of the input polynomials whose non-vanishing is a necessary and sufficient condition for this set of monomials to be a basis of the ring of polynomials in n variables modulo the ideal generated by the system of polynomials. This approach allows us to clarify the algorithms for the Bezout construction of the resultant.

On the Height of the Sylvester Resultant

Let n be a positive integer. We consider the Sylvester resultant of f and g, where f is a generic polynomial of degree 2 or 3 and g is a generic polynomial of degree n. If f is a quadratic polynomial, we find the resultants height. If f is a cubic polynomial, we find tight asymptotics for the resultants height.