Nonlinear Algebra and Applications

TL;DR

The paper surveys nonlinear algebra as a unifying toolkit across eight domains, from polynomial optimization and algebraic statistics to integrable systems, rigidity theory, chemical reaction networks, algebraic vision, and tensor decompositions. It emphasizes a blend of theory and computation—KKT conditions and algebraic degree formulas in optimization, $D$-module theory and holonomic functions in PDEs, toric and Gaussian models in statistics, algebro-geometric KP solutions, and identifiability in tensor decompositions—each supported by modern symbolic-numeric methods. Key contributions include polar-degree refinements of algebraic degrees, Bernstein–Sato frameworks for MLE, Dubrovin threefold constructions for KP solutions, and practical criteria for multistationarity, global rigidity, essential-matrix solvers, and CPD/BTD identifiability. Together these insights demonstrate how nonlinear algebra provides deep structural understanding and robust computational approaches for complex, real-world problems, with broad implications for optimization, statistics, geometry, and vision.

Abstract

We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.

Figures (7)

• Figure 1: Staged tree DMS21 modeling the discrete statistical experiment of flipping a biased coin twice.
• Figure 2: Left: Trott curve. Right: The wave derived from the Trott curve whose parameters are \ref{['eq:pointondubrovin']} at $t=0$CelAgoStu.
• Figure 3: Left: A soliton wave that is taken in Nuevo Vallarta, Mexico by Ablowitz AblBalAbl. Right: A Y-soliton.
• Figure 4: Configuration space of a rhombus in the plane. Left: Placements of the graph in the plane with the bottom two vertices fixed in place and the top two vertices free to move. Right: Projection of the configuration space onto a random three-dimensional subspace. The marked points on the right correspond to blue colored placements on the left, giving two ways to visualize the same data. There are only three singular points since four of the apparent intersections are artifacts of our 2d drawing of a 3d image.
• Figure 5: The left and right images are color-coded to match. Left: We project $401$ points $p^{(i)} \in \mathcal{C}$ onto a random two-dimensional subspace of $\mathbb{R}^{16}.$$200$ orange $\to$ red points approach the singular point $p^{(i)} \to p^\star$ along one branch of the cusp, and another $200$ light-blue $\to$ blue points $p^{(j)} \to p^\star$ approach along the other branch. Right: We view each point $p^{(i)}$ as a placement map $p^{(i)}:V \to \mathbb{R}^2$ sending eleven vertices to the plane, rather than as points $p^{(i)} \in \mathbb{R}^{16}.$ Vertices $1,6,$ and $11$ are pinned and immobile. Right Top: Singular placement $p^\star.$ Right Middle: $200$ light-blue $\to$ blue placements $p^{(i)} \to p^\star$ moving toward the singular placement $p^\star$ along one branch of the cusp. Right Bottom: $200$ orange $\to$ red placements moving toward the singular placement $p^\star$ along the other branch.

Theorems & Definitions (10)

• Definition 1.1
• Theorem 1.2: NR
• Theorem 1.3: DHOST, CJMSV
• Theorem 3.3: diaconis1998algebraic
• Definition 6.1
• Theorem 6.2: FeliuPlos
• Proposition 7.1: Demazure
• Definition 8.1: CPD of tensors
• Definition 8.2: BTD of tensors
• Lemma 8.3: QPL2016