Section: New Results

Solving Systems over the Reals and Applications

Univariate real root isolation in an extension field and applications

In [11] we present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $B_{\alpha }\in L\left[y\right]$, where $L=\mathbb{Q}\left(\alpha \right)$ is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of ${\widetilde{𝒪}}_B\left(N^8\right)$ for isolating the real roots of $B_{\alpha }$, where $N$ is an upper bound on all the quantities (degree and bitsize) of the input polynomials. The bound becomes ${\widetilde{𝒪}}_B\left(N^7\right)$ if we use Pan's algorithm for isolating the real roots. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of $\mathrm{𝚍𝚎𝚜𝚌𝚊𝚛𝚝𝚎𝚜}$' algorithm. For the former we prove a Boolean complexity bound of ${\widetilde{𝒪}}_B\left(N^{12}\right)$ and for the latter a bound of ${\widetilde{𝒪}}_B\left(N^5\right)$. We present aggregate separation bounds and complexity results for isolating the real roots of all polynomials $B_{{\alpha }_k}$, when ${\alpha }_k$ runs over all the real conjugates of $\alpha$. We show that we can isolate the real roots of all polynomials in ${\widetilde{𝒪}}_B\left(N^5\right)$. Finally, we implemented the algorithms in $𝙲$ as part of the core library of MATHEMATICA and we illustrate their efficiency over various data sets.

On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs

The number of embeddings of minimally rigid graphs in ${ℝ}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work in [13] considers the maximal number of real embeddings of minimally rigid graphs in ${ℝ}^3$. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the a priori number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in ${ℝ}^3$, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in ${ℝ}^3$.

Lower bounds on the number of realizations of rigid graphs

Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Towards this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential. Combining computational results with the theory of constructing new rigid graphs by gluing, in [4] we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gröbner basis computations.

The Complexity of Subdivision for Diameter-Distance Tests

In [1] we present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call such tests diameter-distance tests. We illustrate that diameter-distance tests are common in the literature by proving that many interval arithmetic-based tests are, in fact, diameter-distance tests. For this class of algorithms, we provide both non-adaptive bounds for the complexity, based on separation bounds, as well as adaptive bounds, by applying the framework of continuous amortization. Using this structure, we provide the first complexity analysis for the algorithm by Plantinga and Vegeter for approximating real implicit curves and surfaces. We present both adaptive and non-adaptive a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and in terms of the bit complexity for the construction. Finally, we construct families of hypersurfaces to prove that our bounds are tight.

Real root finding for equivariant semi-algebraic systems

Let $R$ be a real closed field. In [19] we consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of s symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d<n$. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most $2d-1$ distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by $s$ polynomials of degree $d$ in time ${\left(sn\right)}^{O\left(d\right)}$. This improves the state-of-the-art which is exponential in $n$. When the variables $x_1,...,x_n$ are quantified and the coefficients of the input system depend on parameters $y_1,...,y_t$, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time ${\left(sn\right)}^{O\left(dt\right)}$.

Exact algorithms for semidefinite programs with degenerate feasible set

Let $A_0,...,A_{n}n$ be $m\times m$ symmetric matrices with entries in $Q$, and let $A\left(x\right)$ be the linear pencil $A_0+x_1{A}_1+\cdots +x_n{A}_n$, where $x=(x1,...,xn)$ are unknowns. The linear matrix inequality (LMI) $A\left(x\right)⪰0$ defines the subset of $R^n$, called spectrahedron, containing all points $x$ such that $A\left(x\right)$ has non-negative eigenvalues. The minimization of linear functions over spectrahedra is called semidefinite programming (SDP). Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for solving SDP are mostly based on the interior point method, assuming some non-degeneracy properties such as the existence of interior points in the admissible set. In [21], we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice but we prove that solving such problems can be done in polynomial time if either $n$ or $m$ is fixed.

A lower bound on the positive semidefinite rank of convex bodies

The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidef-inite formulation. In [3] we show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{logd}$ where $d$ is the smallest degree of a polynomial that vanishes on the boundary of the polar of $C$. This improves on the existing bound which relies on results from quantifier elimination. Our proof relies on the Bézout bound applied to the Karush-Kuhn-Tucker conditions of optimality. We discuss the connection with the algebraic degree of semidefinite programming and show that the bound is tight (up to constant factor) for random spectrahedra of suitable dimension.

On the complexity of computing real radicals of polynomial systems

Let $f=(f_1,...,f_s)$ be a sequence of polynomials in $Q[X_1,...,X_n]$ of maximal degree $D$ and $V\subset C^n$ be the algebraic set defined by $f$ and $r$ be its dimension. The real radical $\sqrt[re]{\langle f\rangle }$ associated to $f$ is the largest ideal which defines the real trace of $V$. In [20] when $V$ is smooth, we show that $\sqrt[re]{\langle f\rangle }$ has a finite set of generators with degrees bounded by V. Moreover, we present a probabilistic algorithm of complexity ${\left(snDn\right)}^{O\left(1\right)}$ to compute the minimal primes of $\sqrt[re]{\langle f\rangle }$ . When $V$ is not smooth, we give a probabilistic algorithm of complexity $s^{O\left(1\right)}{\left(nD\right)}^{O\left(nr{2}^r\right)}$ to compute rational parametrizations for all irreducible components of the real algebraic set $V\cap R^n$. Experiments are given to show the efficiency of our approaches.

Algorithms for Weighted Sums of Squares Decomposition of Non-negative Univariate Polynomials

It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a weighted sum of finitely many squares instead of a sum of two squares, then one can choose all coefficients in the representation to lie in the field generated by the coefficients of the polynomial. In particular, this allows an effective treatment of polynomials with rational coefficients. In [9], we describe, analyze and compare both from the theoretical and practical points of view, two algorithms computing such a weighted sums of squares decomposition for univariate polynomials with rational coefficients. The first algorithm, due to the third author relies on real root isolation, quadratic approximations of positive polynomials and square-free decomposition but its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm. They are exponential in the degree of the input univariate polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using quantifier elimination and root isolation bounds. The second algorithm, due to Chevillard, Harrison, Joldes and Lauter, relies on complex root isolation and square-free decomposition and has been introduced for certifying positiveness of poly-nomials in the context of computer arithmetics. Again, its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm, which are polynomial in the degree of the input polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using Vieta's formula and root isolation bounds. Finally, we report on our implementations of both algorithms and compare them in practice on several application benchmarks. While the second algorithm is, as expected from the complexity result, more efficient on most of examples, we exhibit families of non-negative polynomials for which the first algorithm is better.

On Exact Polya and Putinar's Representations

We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. In [18] we start by providing a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. Next, we apply this algorithm to compute exact Polya and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also compare the implementation of our algorithms with existing methods in computer algebra including cylindrical algebraic decomposition and critical point method.