Compact Formulae in Sparse Elimination

It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-B{'e}zout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.

💡 Research Summary

The paper surveys a collection of compact formulae that arise in the theory of sparse (toric) elimination, emphasizing how the Newton polytope of a polynomial can be exploited to obtain concise expressions for root counts, mixed volumes, resultants, discriminants, and implicit equations.

First, the authors revisit the m‑Bézout bound for systems whose variables are partitioned into S blocks of sizes n₁,…,n_S. By encoding the block‑wise degrees a_{ij} in an S×S matrix A, they show that the bound equals the coefficient of x₁^{n₁}…x_S^{n_S} in the multivariate expansion of 1/ det(I−VA), where V=diag(x₁,…,x_S). This follows from MacMahon’s Master Theorem and yields a generating function for the m‑Bézout number. In the special case of totally mixed Nash equilibrium (TMNE) systems, the matrix A has zeros on the diagonal and ones elsewhere, leading to a closed form 1/(1−σ₂−2σ₃−…−(S−1)σ_S), with σ_k the elementary symmetric polynomials.

Next, the paper addresses the toric root bound (the BKK bound) expressed as the mixed volume MV(Q₁,…,Q_N) of the Newton polytopes. For multihomogeneous systems where each block’s Newton polytope is a scalar multiple of a fixed polytope Γ_j, the mixed volume can be written as perm(A)·∏_{j=1}^S vol(Γ_j) divided by the factorial product n₁!…n_S!. Thus the mixed volume reduces to a matrix permanent, a #P‑complete quantity, but one that can be computed efficiently for many structured instances.

The bulk of the survey concerns sparse resultants. Starting from Macaulay’s classical determinant formula for dense systems, the authors trace the development of determinantal and rational formulae for sparse systems: Dixon’s formulation, D’Andrea’s Macaulay‑style construction, and hybrid Sylvester/Bézout matrices for multihomogeneous and multigraded systems. They highlight the precise conditions under which pure Bézout‑type matrices exist (equivalently, Sylvester‑type matrices exist) and discuss the role of scaled Newton polytopes, toric Jacobians, and hybrid constructions that combine blocks of Sylvester and Bézout rows. For unmixed systems of three dense polynomials, they cite Eisenbud–Schreyer’s Pfaffian formulas and subsequent extensions to higher degrees and numbers of equations.

Discriminants are treated as the natural companion of resultants. While a general compact discriminant formula is absent, the authors present a new 6×6 Sylvester‑type determinantal expression for the discriminant of a specific multilinear system that models TMNE in a three‑player, two‑strategy game. The matrix entries are either zero or the original coefficients, and its determinant vanishes exactly when the system has a multiple root, thus providing an explicit discriminant for this important class of sparse systems.

Finally, the paper proposes an alternative notion of a compact formula: the Newton polytope of the unknown polynomial (resultant, discriminant, or implicit equation). They argue that computing this polytope is often more tractable than expanding the full polynomial. Using an oracle‑based, output‑sensitive algorithm, one can obtain the resultant polytope, then build an interpolation matrix whose rows correspond to monomials in the polytope. This matrix enables efficient geometric operations (intersection, projection, etc.) without ever materializing the full implicit equation. The authors also discuss syzygy‑based implicitization, μ‑bases, and moving curves/surfaces, showing how sparse elimination can be leveraged to compute these structures for parametrized varieties, point‑cloud models, and higher‑codimension objects.

Overall, the survey unifies a wide range of compact expressions—generating functions, permanents, determinant and Pfaffian formulas, and Newton‑polytope based representations—demonstrating their theoretical significance and practical utility in solving and analyzing sparse polynomial systems.