A sparse overview on sparse resultants

TL;DR

This survey surveys advances in sparse resultant theory and computation, focusing on how supports $\mathcal{A}_i$ and Newton polytopes drive elimination and implicitization. It contrasts the Canny–Emiris rational determinant formula with a toric Koszul-determinant approach, highlighting the roles of mixed subdivisions, row content, and the Cayley trick. It also discusses methods to compute the Newton polytope of the resultant and its projections via oracle-based algorithms, linking combinatorial geometry, toric geometry, and computational algebra to enable efficient sparse elimination. Overall, the work connects geometric and algebraic viewpoints to advance elimination and implicitization in sparse polynomial systems.

Abstract

In this survey, we give an overview of advances in the theory and computation of sparse resultants. First, we examine the construction and proof of the Canny-Emiris formula, which gives a rational determinantal formula. Second, we discuss and compare the latter with the computation of the sparse resultant as the determinant of the Koszul complex given by $n + 1$ nef divisors in a toric variety. Finally, we cover techniques for computing the Newton polytope of sparse resultants.

Figures (6)

• Figure 1: Newton polytope of a degree-$5$ polynomial in two variables. Each lattice point corresponds to a monomial. The six monomials in red correspond to the vertices. The dashed triangle represents the Newton polytope of a dense polynomial of the same degree.
• Figure 2: The way to construct coherent mixed subdivisions is considering piecewise affine convex lifting functions $\rho_i: \Delta_i \xrightarrow[]{} \mathbb{R}$. Then, take the Minkowski sum of the graphs of these functions as polytopes in $\mathbb{R}^{n+1}$ and project the lower convex hull back to $\mathbb{R}^n$. Each of the faces of the lower convex hull defines a cell in the subdivision.
• Figure 3: The normal fans of a triangle and a square in $\mathbb{R}^2$. They define the complete toric varieties $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$, respectively.
• Figure 4: The Cayley trick applied to a triangle and a square.
• Figure 5: (a) The secondary polytope $\mathop{\mathrm{\Sigma-pol}}\nolimits(\mathop{\mathrm{Cay}}\nolimits(\mathcal{A}))$ of two triangles (dark, light grey) and one segment $\mathcal{A}_0=\{(0,0),(1,2),(4,1)\},\,\mathcal{A}_1=\{(0,1),(1,0)\},\,\mathcal{A}_2=\{(0,0),(0,1),(2,0)\}$, where $\mathop{\mathrm{Cay}}\nolimits(\mathcal{A})$ is defined as in \ref{['EQ:Cayley']}; its vertices correspond to mixed subdivisions of the Minkowski sum $\mathcal{A}_0+\mathcal{A}_1+\mathcal{A}_2$ and edges to flips between them (b) $N(\mathop{\mathrm{Res_{\mathcal{A}}}}\nolimits)$, whose vertices correspond to the dashed classes of $\mathop{\mathrm{\Sigma-pol}}\nolimits(\mathop{\mathrm{Cay}}\nolimits(\mathcal{A}))$. Bold edges of $\mathop{\mathrm{\Sigma-pol}}\nolimits(\mathop{\mathrm{Cay}}\nolimits(\mathcal{A}))$, called cubical flips, map to edges of $N(\mathop{\mathrm{Res}}\nolimits_{\mathcal{A}})$ (c) $4$-dimensional $N(\mathop{\mathrm{Res}}\nolimits_{\mathcal{A}})$ of 3 generic trinomials.

Theorems & Definitions (31)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Theorem 2.4
• Example 2.5
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Remark 3.4
• Example 3.5