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The Canny-Emiris conjecture for the sparse resultant
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We present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.
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D'ANDREA, Carlos, JERONIMO, Gabriela i SOMBRA, Martín. The Canny-Emiris conjecture for the sparse resultant. Foundations of Computational Mathematics. 2022. ISSN 1615-3375. [consulta: 7 de maig de 2026]. Disponible a: https://hdl.handle.net/2445/193831