Semi-algebraic Sets Research Articles

Describing amoebas

An amoeba is the image of a subvariety of an algebraic torus under the logarithmic moment map. We consider some qualitative aspects of amoebas, establishing some results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semi-algebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba that is not of full dimension is not such a finite intersection if its variety is nondegenerate and we describe amoebas of lines as explicit semi-algebraic sets.

T-orbit spaces of multiplicative actions and applications

We present the results of our recent article [4] and discuss its applications [5]. A finite group with an integer representation has a multiplicative action on the ring of Laurent polynomials, which is induced by a nonlinear action on the compact torus. We study the structure of the orbit space as the image of the fundamental invariants. For the Weyl groups associated to crystallographic root systems of types A, B, C, D, this image is a compact basic semi-algebraic set. We give the defining polynomial inequalities explicitly as the positivity-locus of a Hermite matrix polynomial. As an application, we consider the problem of computing the optimal value of an exponential function and solve it with algebraic methods under symmetry assumptions.

On the Existence of Global Minima and Convergence Analyses for Gradient Descent Methods in the Training of Deep Neural Networks

In this article we study fully-connected feedforward deep ReLU ANNs with an arbitrarily large number of hidden layers and we prove convergence of the risk of the GD optimization method with random initializations in the training of such ANNs under the assumption that the unnormalized probability density function of the probability distribution of the input data of the considered supervised learning problem is piecewise polynomial, under the assumption that the target function (describing the relationship between input data and the output data) is piecewise polynomial, and under the assumption that the risk function of the considered supervised learning problem admits at least one regular global minimum. In addition, in the special situation of shallow ANNs with just one hidden layer and one-dimensional input we also verify this assumption by proving in the training of such shallow ANNs that for every Lipschitz continuous target function there exists a global minimum in the risk landscape. Finally, in the training of deep ANNs with ReLU activation we also study solutions of gradient flow (GF) differential equations and we prove that every non-divergent GF trajectory converges with a polynomial rate of convergence to a critical point (in the sense of limiting Fr\'echet subdifferentiability). Our mathematical convergence analysis builds up on ideas from our previous article Eberle et al., on tools from real algebraic geometry such as the concept of semi-algebraic functions and generalized Kurdyka-Lojasiewicz inequalities, on tools from functional analysis such as the Arzel\`a-Ascoli theorem, on tools from nonsmooth analysis such as the concept of limiting Fr\'echet subgradients, as well as on the fact that the set of realization functions of shallow ReLU ANNs with fixed architecture forms a closed subset of the set of continuous functions revealed by Petersen et al.

Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings

Given a closed semi-algebraic set X ⊂ R n X \subset \mathbb {R}^n and a continuous semi-algebraic mapping G : X → R m G \colon X \to \mathbb {R}^m , it will be shown that there exists an open dense semi-algebraic subset U \mathscr {U} of L ( R n , R m ) L(\mathbb {R}^n, \mathbb {R}^m) , the space of all linear mappings from R n \mathbb {R}^n to R m \mathbb {R}^m , such that for all F ∈ U F \in \mathscr {U} , the image ( F + G ) ( X ) (F + G)(X) is a closed (semi-algebraic) set in R m \mathbb {R}^m . To do this, we study the tangent cone at infinity C ∞ X C_\infty X and the set E ∞ X ⊂ C ∞ X E_\infty X \subset C_\infty X of (unit) exceptional directions at infinity of X X . Specifically we show that the set E ∞ X E_\infty X is nowhere dense in C ∞ X ∩ S n − 1 C_\infty X \cap \mathbb {S}^{n - 1} .

On the complexity of Putinar–Vasilescu's Positivstellensatz

We provide a new degree bound on the weighted sum-of-squares (SOS) polynomials for Putinar–Vasilescu's Positivstellensatz. This leads to another Positivstellensatz saying that if f is a polynomial of degree at most 2df nonnegative on a semialgebraic set having nonempty interior defined by finitely many polynomial inequalities gj(x)≥0, j=1,…,m with g1:=L−‖x‖22 for some L>0, then there exist positive constants c¯ and c depending on f,gj such that for any ε>0, for all k≥c¯ε−c, f has the decomposition(1)(1+‖x‖22)k(f+ε)=σ0+∑j=1mσjgj, for some SOS polynomials σj being such that the degrees of σ0,σjgj are at most 2(df+k). Here ‖⋅‖2 denotes the ℓ2 vector norm. As a consequence, we obtain a converging hierarchy of semidefinite relaxations for lower bounds in polynomial optimization on basic compact semialgebraic sets. The complexity of this hierarchy is O(ε−c) for prescribed accuracy ε>0. In particular, if m=L=1 then c=65, yielding the complexity O(ε−65) for the minimization of a polynomial on the unit ball.

Infinite Energy Quasi-Periodic Solutions to Nonlinear Schrödinger Equations on ℝ

Abstract We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schrödinger equations on $\mathbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous results [3, 4], and the generalization [32], are quasi-periodic in time, but periodic in space. This paper generalizes J. Bourgain’s [5] semi-algebraic set method to analyze nonlinear PDEs, in the non-compact space quasi-periodic setting on $\mathbb R$.

Existence of Two View Chiral Reconstructions

A fundamental question in computer vision is whether a set of point pairs is the image of a scene that lies in front of two cameras. Such a scene and the cameras together are known as a chiral reconstruction of the point pairs. In this paper, we provide a complete classification of $k$ point pairs for which a chiral reconstruction exists. The existence of chiral reconstructions is equivalent to the nonemptiness of certain semialgebraic sets. We describe these sets and develop tools to certify their nonemptiness. For up to three point pairs, we prove that a chiral reconstruction always exists while the set of five or more point pairs that do not have a chiral reconstruction is Zariski-dense. We show that for five generic point pairs, the chiral region is bounded by line segments in a Schläfli double six on a cubic surface with 27 real lines. Four point pairs have a chiral reconstruction unless they belong to two nongeneric combinatorial types, in which case they may or may not.

Characterizing positively invariant sets: Inductive and topological methods

We present two characterizations of positive invariance of sets under the flow of systems of ordinary differential equations. The first characterization uses inward sets which intuitively collect those points from which the flow evolves within the set for a short period of time, whereas the second characterization uses the notion of exit sets, which intuitively collect those points from which the flow immediately leaves the set. Our proofs emphasize the use of the real induction principle as a generic and unifying proof technique that captures the essence of the formal reasoning justifying our results and provides cleaner alternative proofs of known results. The two characterizations presented in this article, while essentially equivalent, lead to two rather different decision procedures (termed respectively LZZ and ESE) for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations. The procedure LZZ improves upon the original work by Liu, Zhan and Zhao (Liu et al., 2011). The procedure ESE, introduced in this article, works by splitting the problem, in a principled way, into simpler sub-problems that are easier to check, and is shown to exhibit substantially better performance compared to LZZ on problems featuring semi-algebraic sets described by formulas with non-trivial Boolean structure.

Intersection bodies of polytopes

We investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for its computation. Moreover, we compute the irreducible components of the algebraic boundary and provide an upper bound for the degree of these components.

Exchangeable and sampling-consistent distributions on rooted binary trees

AbstractWe introduce a notion of finite sampling consistency for phylogenetic trees and show that the set of finitely sampling-consistent and exchangeable distributions on n-leaf phylogenetic trees is a polytope. We use this polytope to show that the set of all exchangeable and sampling-consistent distributions on four-leaf phylogenetic trees is exactly Aldous’ beta-splitting model, and give a description of some of the vertices for the polytope of distributions on five leaves. We also introduce a new semialgebraic set of exchangeable and sampling consistent models we call the multinomial model and use it to characterize the set of exchangeable and sampling-consistent distributions. Using this new model, we obtain a finite de Finetti-type theorem for rooted binary trees in the style of Diaconis’ theorem on finite exchangeable sequences.

Functional norms, condition numbers and numerical algorithms in algebraic geometry

Abstract In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand to optimise accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty }$ . This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty }$ -norm, the use of $L_p$ -norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale’s 17th problem).

Solving parametric systems of polynomial equations over the reals through Hermite matrices

We design a new algorithm for solving parametric systems of equations having finitely many complex solutions for generic values of the parameters. More precisely, let f=(f1,…,fm)⊂Q[y][x] with y=(y1,…,yt) and x=(x1,…,xn), V⊂Ct×Cn be the algebraic set defined by the simultaneous vanishing of the fi's and π be the projection (y,x)→y. Under the assumptions that f admits finitely many complex solutions when specializing y to generic values and that the ideal generated by f is radical, we solve the following algorithmic problem. On input f, we compute semi-algebraic formulas defining open semi-algebraic sets S1,…,Sℓ in the parameters' space Rt such that ∪i=1ℓSi is dense in Rt and, for 1≤i≤ℓ, the number of real points in V∩π−1(η) is invariant when η ranges over Si.This algorithm exploits special properties of some well chosen monomial bases in the quotient algebra Q(y)[x]/I where I⊂Q(y)[x] is the ideal generated by f in Q(y)[x] as well as the specialization property of the so-called Hermite matrices which represent Hermite's quadratic forms. This allows us to obtain “compact” representations of the semi-algebraic sets Si by means of semi-algebraic formulas encoding the signature of a given symmetric matrix.When f satisfies extra genericity assumptions (such as regularity), we use the theory of Gröbner bases to derive complexity bounds both on the number of arithmetic operations in Q and the degree of the output polynomials. More precisely, letting d be the maximal degrees of the fi's and D=n(d−1)dn, we prove that, on a generic input f=(f1,…,fn), one can compute those semi-algebraic formulas using O˜((t+Dt)23tn2t+1d3nt+2(n+t)+1) arithmetic operations in Q and that the polynomials involved in these formulas have degree bounded by D.We report on practical experiments which illustrate the efficiency of this algorithm, both on generic parametric systems and parametric systems coming from applications since it allows us to solve systems which were out of reach on the current state-of-the-art.

Positive Polynomials and Boundary Interpolation with Finite Blaschke Products

The famous Jones–Ruscheweyh theorem states that n distinct points on the unit circle can be mapped to n arbitrary points on the unit circle by a Blaschke product of degree at most n-1. In this paper, we provide a new proof using real algebraic techniques. First, the interpolation conditions are rewritten into complex equations. These complex equations are transformed into a system of polynomial equations with real coefficients. This step leads to a “geometric representation” of Blaschke products. Then another set of transformations is applied to reveal some structure of the equations. Finally, the following two fundamental tools are used: a Positivstellensatz by Prestel and Delzell describing positive polynomials on compact semialgebraic sets using Archimedean module of length N. The other tool is a representation of positive polynomials in a specific form due to Berr and Wörmann. This, combined with a careful calculation of leading terms of occurring polynomials finishes the proof.

The Art Gallery Problem is ∃ℝ-complete

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.

Sum-of-squares chordal decomposition of polynomial matrix inequalities

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all xin mathbb {R}^n if and only if there exists a sum-of-squares (SOS) polynomial sigma (x) such that sigma P is a sum of sparse SOS matrices. Second, we show that setting sigma (x)=(x_1^2 + cdots + x_n^2)^nu for some integer nu suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set mathcal {K}={x:g_1(x)ge 0,ldots ,g_m(x)ge 0} satisfying the Archimedean condition, then P(x) = S_0(x) + g_1(x)S_1(x) + cdots + g_m(x)S_m(x) for matrices S_i(x) that are sums of sparse SOS matrices. Finally, if mathcal {K} is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for (x_1^2 + cdots + x_n^2)^nu P(x) with some integer nu ge 0 when P and g_1,ldots ,g_m are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.

Wilson loops in SYM $\mathcal{N}=4$ do not parametrize an orientable space

We explore the geometric space parametrized by (tree level) Wilson loops in SYM \mathcal{N}=4 . We show that this space can be seen as a vector bundle over a totally non-negative subspace of the Grassmannian, \mathcal{W}_{k,n} . Furthermore, we explicitly show that this bundle is non-orientable in the majority of the cases, and conjecture that it is non-orientable in the remaining situation. Using the combinatorics of the Deodhar decomposition of the Grassmannian, we identify subspaces \Sigma(W)\subset\mathcal{W}_{k,n} for which the restricted bundle lies outside the positive Grassmannian. Finally, while probing the combinatorics of the Deodhar decomposition, we give a diagrammatic algorithm for reading equations determining each Deodhar component as a semialgebraic set.

Quantitative curve selection lemma

We prove a quantitative version of the curve selection lemma. Denoting by s, d, k bounds the number, the maximum total degree and the number of variables of the polynomials describing a semi-algebraic set S and a point x in \({{\bar{S}}}\), we find a semi-algebraic path starting at x and entering in S with a description of degree \((O(d)^{3k+3},O(d)^{k})\) (using a precise definition of the description of a semi-algebraic path and its degree given in the paper). As a consequence, we prove that there exists a semi-algebraic path starting at x and entering in S, such that the degree of the Zariski closure of the image of this path is bounded by \(O(d)^{4k+3}\), improving a result in Jelonek and Kurdyka (Math Z 276:557–570, 2014). We also give an algorithm for describing the real isolated points of S whose complexity is bounded by \(s^{2 k+1}d^{O(k)}\) improving a result in Le et al. (Computing the real isolated points of an algebraic hypersurface, 2020).

A study of the double pendulum using polynomial optimization.

In dynamical systems governed by differential equations, a guarantee that trajectories emanating from a given set of initial conditions do not enter another given set can be obtained by constructing a barrier function that satisfies certain inequalities on the phase space. Often, these inequalities amount to nonnegativity of polynomials and can be enforced using sum-of-squares conditions, in which case barrier functions can be constructed computationally using convex optimization over polynomials. To study how well such computations can characterize sets of initial conditions in a chaotic system, we use the undamped double pendulum as an example and ask which stationary initial positions do not lead to flipping of the pendulum within a chosen time window. Computations give semialgebraic sets that are close inner approximations to the fractal set of all such initial positions.

Design and analysis of a synthetic prediction market using dynamic convex sets

We present a synthetic prediction market whose agent purchase logic is defined using a sigmoid transformation of a convex semi-algebraic set defined in feature space. Asset prices are determined by a logarithmic scoring market rule. Time varying asset prices affect the structure of the semi-algebraic sets leading to time-varying agent purchase rules. We show that under certain assumptions on the underlying geometry, the resulting synthetic prediction market can be used to arbitrarily closely approximate a binary function defined on a set of input data. We also provide sufficient conditions for market convergence and show that under certain instances markets can exhibit limit cycles in asset spot price. We provide an evolutionary algorithm for training agent parameters to allow a market to model the distribution of a given data set and illustrate the market approximation using three open source data sets. Results are compared to standard machine learning methods.

Center problem and ν-cyclicity of polynomial zero-Hopf singularities with non-singular rotation axis

We consider three-dimensional polynomial families of vector fields parameterized by the admissible coefficients having a fixed zero-Hopf equilibrium and a non-singular rotation axis through it. We are interested in the periodic ν -orbits, that is, those orbits that makes a fixed arbitrary number ν (or a divisor of ν ) of revolutions about the rotation axis and then returns to the initial point closing the orbit. We develop a Bautin-type method to study the ν -cyclicity of the equilibrium, that is, the maximum number of small-amplitude ν -limit cycles (isolated periodic ν -orbits) that can be made to bifurcate from the equilibrium by moving the parameters of the family restricted to some open semi-algebraic sets. The method uses branching theory based on Newton-Puiseux Theorem to get a finite number of an analytic reduced one-dimensional Poincaré maps with associated Bautin ideal on certain Noetherian ring of rational functions on an extended parameter space. We derive global upper bounds on the number of bifurcated ν -limit cycles even in the infinite codimension case for which the perturbation of a local two-dimensional invariant manifold through the singularity completely foliated by periodic ν -orbits needs to be performed. A cubic normal form serves as an example of this procedure.