Conditions for Exact Resultants using the Dixon Formulation

Proc. of International Symposium on Symbolic and Algebraic Computation (ISSAC),
August 2000, St. Andrews, Scotland.

Arthur Chtcherba and Deepak Kapur

A structural criteria on multivariate polynomial systems is developed such that the generalized Dixon formulation of multivariate resultants defined by Kapur, Saxena and Yang (ISSAC, 1994) computes the resultant exactly, i.e. no extraneous factors are produced in the resultant computations for such polynomial systems. The concept of a Dixon-exact support is introduced; it is proved that the Dixon resultant formulation produces the exact resultant for generic unmixed systems whose support is Dixon-exact. A geometric operation, called direct-sum, on polytopes is defined that preserves the property of supports being Dixon-exact. Generic n-degree systems for which the Dixon formulation is known to compute exact resultants are shown to be a special case of generic unmixed polynomial systems whose support is Dixon-exact. Multigraded systems introduced by Strumfels and Zelvinsky for which they gave a Sylvester type formula for resultants are also shown to be a special case of generic unmixed polynomial systems whose support is Dixon-exact. Using the techniques discussed in a paper by Kapur and Saxena (ISSAC, 1997), a wide class of polynomial systems can be identified for which the Dixon formulation produces exact resultants.

For the bivariate case, an exhaustive analysis of monomials in a polynomial system is given vis a vis their role in producing extraneous factors in a projection operator computed using the generalized Dixon formulation. Such an analysis is likely to give insights for the general case of elimination of arbitrarily many variables.

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