arXiv subjects

Rainer Sinn

Publications and source records attributed to Rainer Sinn.

At least 19 recordsLinked to original sources

arXiv2026

Local dimension testing via quadratic obstructions

We give an algorithm, based on second-order necessary conditions, to test whether a point is isolated on an algebraic set. Using this subroutine, we develop an algorithm that bounds the local dimension of an algebraic set at a point. The principal computation in our isolation test is governed by the nullity of the Jacobian, avoiding the combinatorial growth of classical methods. For exact input, the bound returned by the algorithm is certified. We compare our approach with classical methods. As an application, we show that the realization space of the 24-cell has the expected local dimension at its symmetric regular realization, completing the local dimension program for regular 4-polytopes initiated by Rastanawi, Sinn, and Ziegler.

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arXiv2026

Stubborn Polynomials

The relationship between nonnegative polynomials and sums of squares is a classical topic in real algebraic geometry. We study \emph{stubborn polynomials} $f$ on a real variety $X$, which are polynomials nonnegative on $X$, such that no odd power of $f$ is a sum of squares. Previously, stubborn polynomials were studied only in the globally nonnegative case, with results restricted to polynomials nonnegative on $\mathbb{P}^2$. We fully characterize stubborn polynomials on smooth curves, showing that a polynomial on a smooth totally real curve is stubborn if and only if all of its zeros are real. This implies that there exist smooth curves with no stubborn polynomials in low degree, while stubborn polynomials must exist in sufficiently high degrees on curves with positive genus. We explore the much more delicate situation with singular and reducible curves. While being real-rooted always implies being stubborn, there also exist singular curves with no stubborn polynomials at all. We then analyze the case of ternary sextics, i.e.,~polynomials of degree $6$ on $\mathbb{P}^2$. We prove the Conjecture of Blekherman, Kozhasov, and Reznick that a nonnegative ternary sextic is stubborn if and only if its real delta-invariant is at least 9. To analyze this case, we develop results for lifting stubborn polynomials from curves to higher dimensional varieties and use the theory of weak Del Pezzo surfaces. We complement these results with structural properties of stubborn polynomials and present many explicit examples.

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arXiv2026

Positive Genus Pairs from Amplituhedra

A main conjecture in the field of Positive Geometry states that amplituhedra, which are certain semi-algebraic sets in the Grassmannian, are positive geometries. It is motivated by examples showing that the canonical forms of certain amplituhedra compute scattering amplitudes in particle physics. Beyond a small number of special cases, this conjecture is still open. In recent work, Brown and Dupont introduced a new framework, based on mixed Hodge theory, connecting canonical forms and de Rham cohomology via genus zero pairs. We give short proofs that the amplituhedron gives rise to a genus zero pair in the cases when it is known to be a positive geometry. However, in the general case we show that amplituhedra inside the Grassmannian give rise to pairs of strictly positive genus. We provide an explicit example of a genus one pair arising from a positive geometry in projective space, showing that having genus zero is not a necessary condition to be a positive geometry. Finally, we show that this positive geometry still gives rise to a genus zero pair in a different ambient variety.

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arXiv2025

Toric extensions of P\'olya's theorem

The classical version of P\'olya's theorem provides a simple method for certifying that a homogeneous polynomial of degree d is strictly copositive, that is, it takes only positive values on the nonnegative real orthant. However, this method might fail to detect copositivity of polynomials that are missing certain degree d monomials. In this paper, we present extensions and converses to P\'olya's theorem for sparse polynomials, using techniques from positive toric geometry. Furthermore, we explore how this method can be used to study the convergence of Feynman integrals in particle physics.

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arXiv2025

Taking the amplituhedron to the limit

The amplituhedron is a semialgebraic set given as the image of the non-negative Grassmannian under a linear map subject to a choice of additional parameters. We define the limit amplituhedron as the limit of amplituhedra by sending one of the parameters, namely the number of particles $n$, to infinity. We study this limit amplituhedron for $m = 2$ and any $k$, relating to the number of negative helcity particles. We determine its algebraic boundary in terms of Chow hypersurfaces. This hypersurface in the Grassmannian is stratified by singularities in terms of higher order secants of the rational normal curve. In conclusion, we show that the limit amplituhedron is a positive geometry with a residual arrangement that is empty.

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arXiv2024

Spurious local minima in nonconvex sum-of-squares optimization

We study spurious second-order stationary points and local minima in a nonconvex low-rank formulation of sum-of-squares optimization on a real variety $X$. We reformulate the problem of finding a spurious local minimum in terms of syzygies of the underlying linear series, and also bring in topological tools to study this problem. When the variety $X$ is of minimal degree, there exist spurious second-order stationary points if and only if both the dimension and the codimension of the variety are greater than one, answering a question by Legat, Yuan, and Parrilo. Moreover, for surfaces of minimal degree, we provide sufficient conditions to exclude points from being spurious local minima. In particular, all second-order stationary points associated with infinite Gram matrices on the Veronese surface, corresponding to ternary quartics, lie on the boundary and can be written as a binary quartic, up to a linear change of coordinates, complementing work by Scheiderer on decompositions of ternary quartics as a sum of three squares. For general varieties of higher degree, we give examples and characterizations of spurious second-order stationary points in the interior, together with a restricted path algorithm that avoids such points with controlled step sizes, and numerical experiment results illustrating the empirical successes on plane cubic curves and Veronese varieties.

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arXiv2024

Pythagoras Numbers for Ternary Forms

We study the Pythagoras numbers $py(3,2d)$ of real ternary forms, defined for each degree $2d$ as the minimal number $r$ such that every degree $2d$ ternary form which is a sum of squares can be written as the sum of at most $r$ squares of degree $d$ forms. Scheiderer showed that $d+1\leq py(3,2d)\leq d+2$. We show that $py(3,2d) = d+1$ for $2d = 8,10,12$. The main technical tool is Diesel's characterization of height 3 Gorenstein algebras.

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arXiv2024

Nonnegativity certificates on real algebraic surfaces

We introduce tools for transferring nonnegativity certificates for global sections between line bundles on real algebraic surfaces. As applications, we improve Hilbert's degree bounds on sum-of-squares multipliers for nonnegative ternary forms, give a complete characterization of nonnegative real forms of del Pezzo surfaces, and establish quadratic upper bounds for the degrees of sum-of-squares multipliers for nonnegative forms on real ruled surfaces.

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arXiv2024

Plane quartics and heptagons

Every polygon with n vertices in the complex projective plane is naturally associated with its adjoint curve of degree n-3. Hence the adjoint of a heptagon is a plane quartic. We prove that a general plane quartic is the adjoint of exactly 864 distinct complex heptagons. This number had been numerically computed by Kohn et al. We use intersection theory and the Scorza correspondence for quartics to show that 864 is an upper bound, complemented by a lower bound obtained through explicit analysis of the famous Klein quartic.

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arXiv2024

Nonnegative Polynomials and Moment Problems on Algebraic Curves

The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative polynomials $\pos$ on a smooth real projective curve $X$. We show that there is a duality between its faces and totally real effective divisors on $X$. This allows us to fully describe the face lattice in case $X$ has genus one. We compute the Carath\'{e}odory number of the dual moment cone $\pos^\vee$ for an elliptic normal curve $X$, which measures the complexity of quadrature rules of measures supported on $X$. Interestingly, the topology of the real locus of $X$ influences the Carath\'{e}odory number of $\pos^\vee$. We apply our results to truncated moment problems on affine cubic curves, where we deduce sharp bounds on the flat extension degree.

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arXiv2024

Adjoints and Canonical Forms of Tree Amplituhedra

We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in ${\rm Gr}(2,4)$ are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.

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arXiv2022

Landau on Chess Tournaments and Google's PageRank

In his first mathematical paper, published in 1895 when he was 18, Edmund Landau suggested a new way to determine the winner of a chess tournament by not simply adding for each player the fixed number of points they would get for each win or draw, but rather by considering the performance of all players in the tournament relative to each other: each player would get more credit for games won or drawn against stronger players. Landau called this "relative Wertbemessung", which translates to relative score. The basic idea from linear algebra behind this scoring system was rediscovered and reused in many contexts since 1895; in particular, it is a central ingredient in Google's PageRank.

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arXiv2022

Convex hulls of surfaces in fourspace

This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces do. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese, Del Pezzo and Bordiga surfaces.

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arXiv2022

Tropical Positivity and Determinantal Varieties

We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. Moreover, in the case square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.

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arXiv2022

Weak and Strong Extremal Biquadratics

We study quasiconvex quadratic forms on $n \times m$ matrices which correspond to nonnegative biquadratic forms in $(n,m)$ variables. We disprove a conjecture stated by Harutyunyan--Milton (Comm. Pure Appl. Math. 70(11), 2017) as well as Harutyunyan--Hovsepyan (Arch. Ration. Mech. Anal. 244, 2022) that extremality in the cone of quasiconvex quadratic forms on $3\times 3$ matrices can follow only from the extremality of the determinant of its acoustic tensor, using previous work by Buckley--\v{S}ivic (Linear Algebra Appl. 598, 2020). Our main result is to establish a conjecture of Harutyunyan--Milton (Comm. Pure Appl. Math. 70(11), 2017) that weak extremal quasiconvex quadratics on $3 \times 3$ matrices are strong extremal. Our main technical ingredient is a generalization of the work of Kunert--Scheiderer on extreme nonnegative ternary sextics (Trans. Amer. Math. Soc. 370(6), 2018). Specifically, we show that a nonnegative ternary sextic, which is not a square, is extremal if and only if its variety (over the complex numbers) is a rational curve and all its singularities are real.

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arXiv2022

Moments, Sums of Squares, and Tropicalization

We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set $S$. The truncated cones of moments of measures supported on the set $S$ is dual to nonnegative polynomials on $S$, while "pseudo-moments" are dual to sums of squares approximations to nonnegative polynomials. We provide explicit combinatorial descriptions of tropicalizations of the moment and pseudo-moment cones, and demonstrate their usefulness in distinguishing between nonnegative polynomials and sums of squares. We give examples that show new limitations of sums of squares approximations of nonnegative polynomials. When the semialgebraic set is defined by binomial inequalites, its moment and pseudo-moment cones are closed under Hadamard product. In this case, their tropicalizations are polyhedral cones that encode all binomial inequalities on the moment and pseudo-moment cones.

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arXiv2021

Adjoints and Canonical Forms of Polypols

Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry. We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics.

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arXiv2021

Do alcoved lattice polytopes have unimodal h*-vector?

We show that h*-vectors of alcoved polytopes P in R^n (of Lie type A) are unimodal if they contain interior lattice points and their facets have lattice distance 1 to the set of interior lattice points. The maximal possible such distance for general alcoved polytopes is shown to be dim(P)-1. A secondary purpose of the paper is to serve as a guide to previous work surrounding unimodality of h*-vectors of alcoved polytopes and related questions.

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