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Nonlinear algebra and applications

The first author is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 445466444. The second author is supported by Turkish Scientific and Technological Research Council (TBİTAK) – TBİTAK 2236, Project number 1119B362000396. The third author is supported by a NSF Mathematical Sciences Postdoctoral Research Fellowship (DMS-2103310.) The seventh author is supported by the Göran Gustafsson foundation

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  • 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.

    Mathematics Subject Classification: Primary: 08-02, 13P25; Secondary: 14Q20.


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  • Figure 1.  Staged tree [78] 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 (13) at t = 0 [11]

    Figure 3.  Left: A soliton wave that is taken in Nuevo Vallarta, Mexico by Ablowitz [3,2]. 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

    Figure 6.  Three rigid bars in black, two elastic cables in green. Left: The elastic framework in a stable configuration. Right: Configuration of the framework after crossing the catastrophe discriminant, depicted in red. The three square vertices are pinned, the cross vertex is controlled, and the two circular vertices are free: their position is found by minimizing energy over the configuration space, which is visualized by the grey, dashed coupler curve. Bottom: the energy function along the coupler curve with the current position depicted in green.

    Figure 7.  A pinhole camera with principal point equal to $ (0, \, 0, \, 1) \in H $ and focal length $ 1. $ The point $ (x,y,z) $ is projected onto the plane $ H. $ The resulting image is the point $ (x/z,y/z,1). $ The dashed line corresponds to the point in $ \mathbb{P}_{\mathbb R}^2 $ which is represented in homogeneous coordinates by $ [x:y:z]$

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