On the relationship between the Dixon-based resultant construction and the supports of polynomial systems

Abstract

Different matrix based resultant formulations use the support of the polynomials in a polynomial system in various ways for setting up resultant matrices for computing resultants. Every formulation suffers, however, from the fact that for most polynomial systems, the output is not a resultant, but rather a nontrivial multiple of the resultant, called a projection operator . It is shown that for the Dixon-based resultant methods, the degree of the projection operator of unmixed polynomial systems is determined by the support hull of the support of the polynomial system. This is similar to the property that the Newton polytope of a support determines the degree of the resultant for toric zeros.The support hull of a given support is similar to its convex hull (Newton polytope) except that instead of the Euclidean distance, the support hull is defined using relative quadrant (octant) position of points. The concept of a support hull interior point with respect to a support is defined. It is shown that for unmixed polynomial systems, generic inclusion of terms corresponding to support hull interior points does not change the size of the Dixon matrix (hence, the degree of the projection operator). The support hull of a support is the closure of the support with respect to support-interior points.The above results are shown to hold both for the generalized Dixon formulation as well as for Sylvester-type Dixon dialytic matrices constructed using the Dixon formulation.It is proved that for an unmixed polynomial system, the size of the Dixon matrix is less than or equal to the Minkowski sum of the alternating sums of the successive projections of the support of the polynomial system. This is a refinement of the result in Kapur and Saxena 1996 about the size of the Dixon matrix of a polynomial system, where it was shown that for the unmixed polynomial system, the size of the Dixon matrix is less than or equal to the Minkowski sum of the successive projection of the support.Many other combinatorial properties of the size of the Dixon matrix and the structure of the Dixon polynomial of a given polynomial system are related to the support hull of the polynomial system and their projections along different dimensions.This research is supported in part by NSF grant nos. CCR-0203051, CDA-9503064 and a grant from the Computer Science Research Institute at Sandia National Labs.

On the efficiency and optimality of Dixon-based resultant methods

“…The main advantage of using the Dixon multiplier matrices over the associated Dixon matrices is (i) in the mixed case, the Dixon multiplier matrices can have resultants as their determinants, whereas the Dixon matrices often have determinants which includes along with the resultants, extraneous factors; further, if the determinant of a Dixon multiplier matrix has an extraneous factor with the resultant, the degree of the extraneous factor is lower than the degree of the extraneous factor appearing in the determinant of the Dixon matrix, (ii) the Dixon multiplier matrices can be stored and computed more efficiently, given that the entries are either zero or the coefficients of the monomials in the polynomials; this is in contrast to the entries of the Dixon matrices which are determinants in the coefficients. This paper is a summary of results; details including proofs of the results reported in this paper can be found in [CK02b] and [CK02c]).…”

Constructing Sylvester-type resultant matrices using the Dixon formulation

“…i.e., the number of rows is minimized In the case of (i), a heuristic for choosing such translation vectors t is described in [CK02a]. For minimizing the size of Dixon multiplier matrices, a slightly modified method is needed as the objective is to minimize the sizes of θ i (m), i.e., Φ i (α, t) for each i (which also involves choosing m).…”

“…The support hull of a given support is similar to the associated convex hull; whereas the later is defined in terms of the shortest Euclidean distance, the support hull is defined using the Manhattan distance. (For a complete description, see [CK02a].) Example: Consider a highly mixed polynomial system (see figure 3):…”

“…In this example, it is 2, 2, 2 , where i th entry in the tuple denotes the degree of extraneous factor in terms of coefficients of f i . Note that in this example, α is chosen from the support hull intersection; in general, good choices for α are always from the support hull as the Dixon matrix construction is invariant under the presence of monomials whose exponent is support hull interior (see [CK02a]).…”

Support hull

“…The following theorem connects the support hull and the Cayley-Dixon construction; the proof is omitted here because of lack of space; an interested reader can consult [4,9] for a detailed proof.…”