The Cayley-Dixon formulation for multivariate resultants have been shown to be efficient (both experimentally and theoretically) for computing resultants by simultaneously eliminating many variables from a polynomial system. In this paper, the behavior of Cayley-Dixon resultant construction and the structure of Dixon matrices is analyzed for composed polynomial systems constructed from a multivariate system in which each variable is substituted by a univariate polynomial in a distinct variable. It is shown that Dixon projection operator (multiple of resultant) of the composed system can be expressed as a power of the resultant of the outer polynomial system multiplied by powers of the leading coefficients of the univariate polynomials substituted for variables in the outer system. A new resultant formula is derived for systems where it is known that the Cayley-Dixon construction does not contain extraneous factors. The derivation of the resultant formula for the composed system unifies all the known related results in the literature. Furthermore, it demonstrates that the resultant of a composed system can be effectively calculated by considering only the resultant of the outer system. Since the complexity of resultant computation is typically determined by the degree (and support) of the polynomial system, resultants of a composed system can be computed much faster by focusing only on the outer system.
- Dixon resultant,
- composed polynomials
Available at: http://works.bepress.com/minimair/9/