Comparison of various multivariate resultant formulations
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
The structure of sparse resultant matrices
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
MARS: a MAPLE/MATLAB/C resultant-based solver
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
A subdivision-based algorithm for the sparse resultant
Journal of the ACM (JACM)
Hybrid sparse resultant matrices for bivariate systems
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Symbolic and numeric methods for exploiting structure in constructing resultant matrices
Journal of Symbolic Computation
Hybrid sparse resultant matrices for bivariate polynomials
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Algebraic and numerical algorithms
Algorithms and theory of computation handbook
Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants
Journal of Symbolic Computation
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Given a system of $n+1$ generic Laurent polynomials, for $i \,=\, 1, \ldots, n+1$, $$\eqlabel(\InputSystem) f_i(\x) \quad = \quad \sum_{q\in \A_i} c_{iq} \,x^q; \qquad q \,=\, (q_1,\ldots,q_n); \qquad \x^q \,=\, x_1^{q_1}x_2^{q_2}\cdots x_n^{q_n}; \eqno(\InputSystem) $$ with (finite) support sets $\A_i \subset L$, where $L$ is some affine lattice isomorphic to $\Z^n$; we consider algorithms for the {\it Newton resultant} $R(f_1,f_2, \ldots, f_{n+1})$. This is the unique (up to sign) irreducible polynomial with coefficients in $\Z$ and monomials in the $c_{iq}$ which determines whether or not system~(\InputSystem) has common roots in the {\it algebraic torus} $(\C-\{0\})^n$. The resultant depends only on the {\it Newton polytopes} $N_i \,:=\, conv(\A_i) \subset \R^n$ of the sets $\A_i$. Our terminology emphasizes the dependence on the combinatorics of the Newton polytopes. The algebraic torus is the natural setting for us because we are interested in the properties of systems of polynomials which are invariant under symmetries of the affine lattice $L$, and translation by $q\in L$ corresponds to multiplication by $x^q$ at the level of polynomials. Since $x^q$ may have negative exponents, we restrict to points none of whose coordinates are zero.