[Paper Review] The Canny-Emiris conjecture for the sparse resultant

Carlos D’Andrea, Gabriela Jeronimo+1 more|arXiv (Cornell University)|Apr 30, 2020

Polynomial and algebraic computation14 references4 citations

TL;DR

This paper proves a generalized version of the Canny-Emiris conjecture for the sparse resultant by introducing a product formula for its initial parts using mixed subdivisions of polytopes. The authors establish that the sparse resultant equals the quotient of the determinant of a Canny-Emiris matrix by one of its principal minors under specific combinatorial conditions, extending Macaulay's formula to the sparse setting and confirming the conjecture in full generality for the first time.

ABSTRACT

Abstract 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.

Motivation & Objective

• To resolve the Canny-Emiris conjecture on computing the sparse resultant as a quotient of determinants of Sylvester-type matrices.
• To generalize Sturmfels’ product formula for initial parts of the sparse resultant to arbitrary families of supports.
• To establish conditions under which the sparse resultant equals the determinant of a Canny-Emiris matrix divided by a principal minor.
• To extend the theory of sparse resultants beyond the essential case to full generality, including non-essential supports and non-full-rank lattices.
• To provide a uniform framework for computing sparse resultants via mixed subdivisions and piecewise affine functions.

Proposed method

• Derive a product formula for the initial parts of the sparse resultant using mixed subdivisions of the Minkowski sum of supports.
• Define Canny-Emiris matrices via convex piecewise affine functions and their inf-convolution, constructing a Sylvester-type square matrix.
• Identify a principal submatrix of the Canny-Emiris matrix corresponding to non-mixed cells in the mixed subdivision.
• Use the compatibility of the construction with restriction to lower-dimensional supports to reduce to simpler cases.
• Apply the theory of toric varieties and elimination theory to relate the sparse resultant to the determinant of the Canny-Emiris matrix.
• Prove that the determinant of the Canny-Emiris matrix is a nonzero multiple of the sparse resultant, and that the quotient by the principal minor yields the resultant under admissibility conditions.

Experimental results

Research questions

• RQ1 Under what conditions does the sparse resultant equal the quotient of the determinant of a Canny-Emiris matrix and one of its principal minors?
• RQ2 How can the initial parts of the sparse resultant be expressed as a product over mixed cells in a mixed subdivision?
• RQ3 What is the precise relationship between the Canny-Emiris matrix and the Macaulay formula in the sparse setting?
• RQ4 How does the construction behave under restriction to sub-supports or lower-dimensional affine spans?
• RQ5 When is the mixed subdivision associated to a family of affine functions admissible, ensuring the validity of the quotient formula?

Key findings

• The paper proves the generalized Canny-Emiris conjecture: the sparse resultant equals the determinant of the Canny-Emiris matrix divided by its principal minor when the associated mixed subdivision is admissible.
• A new product formula for the initial parts of the sparse resultant is established, generalizing Sturmfels’ earlier result to arbitrary families of supports.
• The Macaulay formula for the homogeneous resultant appears as a special case of the main result, providing an independent proof for it.
• The construction is extended to non-essential supports and non-full-rank lattices, ensuring uniform behavior across all cases.
• The formula fails when the mixed subdivision is not admissible, as demonstrated by a counterexample with a tight, non-admissible subdivision where the principal minor does not divide the determinant.
• The Canny-Emiris matrix and its principal minor are shown to be compatible with restriction to sub-supports, enabling recursive and inductive arguments.