On the Bézout Construction of the Resultant

Abstract

This paper deals with some ideas of Bézout and his successors Poisson, Netto and Laurent for solving polynomial systems. We treat them from the determinantal and from the Gröbner basis point of view. This results in effective algorithms for constructing the multivariate resultant. Other problems of Elimination Theory are discussed: how to find an eliminant for a polynomial system, how to represent its zeroes as the rational functions of the roots of this eliminant and how to separate zeroes.

Rectangular Corner Cutting and Dixon A -resultants

“…For more discussion on classical resultants and A-resultants, we refer the reader to Bikker and Uteshev (1999), Cox et al (1998), Emiris and Mourrain (1999), Gelfand et al (1994), Kapranov et al (1992) and Saxena (1997).…”

Exact, efficient, and complete arrangement computation for cubic curves

“…3 We shall give two further references: The (sub)resultant property of (sub)determinants of the Bézout matrix is derived from first principles (i.e., without reference to the Sylvester matrix) by Goldman et al [41]. The general Bézout construction of resultants to eliminate several variables at once is treated extensively by Bikker and Uteshev [12], including an explicit discussion of the subresultant properties in the univariate case considered here. 4 …”

“…While quite interesting in general, this would not gain a lot in our setting where m 3, k 1 4. Be warned that the matrix B considered in[12] differs from our Bezoutiant Bez(f, g) by a change of basis 5. The designation "Uspensky's Method" is common but incorrect[3].…”

Metric problems for quadrics in multidimensional space

“…Theoretical possibility of such a representation as well as constructive algorithms for its implementation are discussed in [1]. We note just only that in case of reducibility, the coefficients b ℓj can be expressed as rational functions of the coefficients of g(X).…”