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Gregorio Malajovich

Publications and source records attributed to Gregorio Malajovich.

At least 19 recordsLinked to original sources
arXiv2026

Global analysis of the Kuramoto flow

Kuramoto's differential equation describes a synchronization process between several harmonic oscillators. It has been used to model biological phenomena such as the synchronization of heart cells, the circadian rhythm, or brain waves. It is also used in power system control. The simplest possible model assumes that all oscillators are identical and connected to each other with equal pairwise attraction. In this paper, we give a full geometric description of its global dynamics in terms of Morse theory and dynamical systems. Most of this description is stable in the sense that it is topologically preserved under small perturbations of the parameters.

math.DS↗
arXiv2025

Complexity of sparse polynomial solving 3: Infinity

A theory of numerical path-following in toric varieties was suggested in two previous papers. The motivation is solving systems of polynomials with real or complex coefficients. When those polynomials are not assumed 'dense', solving them over projective space or complex space may introduce spurious, degenerate roots or components. Spurious roots may be avoided by solving over toric varieties. In this paper, a homotopy algorithm is locally defined on charts of the toric variety. Its complexity is bounded linearly by the condition length, that is the integral along the lifted path (coefficients and solution) of thetoric condition number. Those charts allow for stable computations near "toric infinity",which was not possible within the technology of the previous papers.

math.AG↗
arXiv2022

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.

math.MG↗
arXiv2022

Complexity of Sparse Polynomial Solving 2: Renormalization

Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations, or sparse systems of exponential sums. The cost of continuation depends on a renormalized condition length, defined as a line integral of the condition number along all the lifted renormalized paths. The theory developed in this paper leads to a continuation algorithm tracking all the solutions between two generic systems with the same structure. The algorithm is randomized, in the sense that it follows a random path between the two systems. The probability of success is one. In order to produce an expected cost bound, several invariants depending solely of the supports of the equations are introduced. For instance, the mixed area is a quermassintegral that generalizes surface area in the same way that mixed volume generalizes ordinary volume. The facet gap measures for each 1-cone in the fan and for each support polytope, how close is the supporting hyperplane to the nearest vertex. Once the supports are fixed, the expected cost depends on the input coefficients solely through two invariants: the renormalized toric condition number and the imbalance of the absolute values of the coefficients. This leads to a non-uniform polynomial complexity bound for polynomial solving in terms of those two invariants.

math.NA↗
arXiv2019

A theory of NP-completeness and ill-conditioning for approximate real computations

We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The theory admits deterministic and nondeterministic polynomial time recognizable problems. We prove that P is not NP in this theory if and only if P is not NP in the BSS theory over the reals. Then we develop a theory with weak and strong approximate computations. This theory is intended to model actual numerical computations that are usually performed in floating point arithmetic. It admits classes P and NP and also an NP-complete problem. We relate the P vs NP question in this new theory to the classical P vs NP problem.

cs.CC↗
arXiv2017

Complexity of sparse polynomial solving: homotopy on toric varieties and the condition metric

This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of $n$-variate polynomial equations is specified through $n$ monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. This variety is a compactification of $(\mathbb C\setminus\{0\})^n$, dependent on the monomial bases. A toric Newton operator is defined on that toric variety. Smale's alpha theory is generalized to provide criteria of quadratic convergence. Two condition numbers are defined and a higher derivative estimate is obtained in this setting. The Newton operator and related condition numbers turn out to be invariant through a group action related to the momentum map. A homotopy algorithm is given, and is proved to terminate after a number of Newton steps which is linear on the condition length of the lifted homotopy path. This generalizes a result from Shub (2009).

math.NA↗
arXiv2016

Computing mixed volume and all mixed cells in quermassintegral time

The mixed volume counts the roots of generic sparse polynomial systems. Mixed cells are used to provide starting systems for homotopy algorithms that can find all those roots, and track no unnecessary path. Up to now, algorithms for that task were of enumerative type, with no general non- exponential complexity bound. A geometric algorithm is introduced in this paper. Its complexity is bounded in the average and probability-one settings in terms of some geometric invariants: quermassintegrals associated to the tuple of convex hulls of the support of each polynomial. Besides the complexity bounds, numerical results are reported. Those are consistent with an output- sensitive running time for each benchmark family where data is available. For some of those families, an asymptotic running time gain over the best code available at this time was noticed.

math.NA↗
arXiv2013

On the expected number of zeros of nonlinear equations

This paper investigates the expected number of complex roots of nonlinear equations. Those equations are assumed to be analytic, and to belong to certain inner product spaces. Those spaces are then endowed with the Gaussian probability distribution. The root count on a given domain is proved to be `additive' with respect to a product operation of functional spaces. This allows to deduce a general theorem relating the expected number of roots for unmixed and mixed systems. Examples of root counts for equations that are not polynomials nor exponential sums are given at the end.

math.AG↗
arXiv2012

Newton iteration, conditioning and zero counting

Those lectures revolve around the following problem: given a system of n real polynomials in n variables, count the number of real roots. The first lecture is a course on Newton iteration and alpha-theory. The second describes an inclusion-exclusion algorithm for real polynomials, developed by Felipe Cucker, Teresa Krick, Mario Wschebor and myself. The third lecture introduces tools for complexity analysis of numerical algorithms, and uses those tools to analyze our root-counting algorithm.

math.NA↗
arXiv2012

Convexity properties of the condition number II

In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the condition metric in the space of maximal rank matrices. Here, we show that this condition metric induces a Lipschitz-Riemann structure on that space. After investigating geodesics in such a nonsmooth structure, we show that the inverse of the smallest singular value of a matrix is a log-convex function along geodesics (Theorem 1). We also show that a similar result holds for the solution variety of linear systems (Theorem 31). Some of our intermediate results, such as Theorem 12, on the second covariant derivative or Hessian of a function with symmetries on a manifold, and Theorem 29 on piecewise self-convex functions, are of independent interest. Those results were motivated by our investigations on the com- plexity of path-following algorithms for solving polynomial systems.

math.DG↗
arXiv2011

Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations

Given a C^1 path of systems of homogeneous polynomial equations f_t, t in [a,b] and an approximation x_a to a zero zeta_a of the initial system f_a, we show how to adaptively choose the step size for a Newton based homotopy method so that we approximate the lifted path (f_t,zeta_t) in the space of (problems, solutions) pairs. The total number of Newton iterations is bounded in terms of the length of the lifted path in the condition metric.

math.NA↗
arXiv2011

A Numerical Algorithm for Zero Counting. III: Randomization and Condition

In a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605, 2008) we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number kappa(f) for the input system f. In this paper, we look at kappa(f) as a random variable derived from imposing a probability measure on the space of polynomial systems and give bounds for both the tail P{kappa(f) > a} and the expected value E(log kappa(f)).

math.NA↗
arXiv2009

Convexity properties of the condition number

We define in the space of n by m matrices of rank n, n less or equal than m, the condition Riemannian structure as follows: For a given matrix A the tangent space of A is equipped with the Hermitian inner product obtained by multiplying the usual Frobenius inner product by the inverse of the square of the smallest singular value of A denoted sigma_n(A). When this smallest singular value has multiplicity 1, the function A -> log (sigma_n(A)^(-2)) is a convex function with respect to the condition Riemannian structure that is t -> log (sigma_n(A(t))^(-2)) is convex, in the usual sense for any geodesic A(t). In a more abstract setting, a function alpha defined on a Riemannian manifold (M,<,>) is said to be self-convex when log alpha (gamma(t)) is convex for any geodesic in (M,<,>). Necessary and sufficient conditions for self-convexity are given when alpha is C^2. When alpha(x) = d(x,N)^(-2) where d(x,N) is the distance from x to a C^2 submanifold N of R^j we prove that alpha is self-convex when restricted to the largest open set of points x where there is a unique closest point in N to x. We also show, using this more general notion, that the square of the condition number ||A|||_F / sigma_n(A) is self-convex in projective space and the solution variety.

math.NA↗
arXiv2008

A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials' degree, and kappa(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a polynomial bound for the precision required to ensure the returned output is correct is exhibited. This bound is a major feature of our algorithm since it is in contrast with the exponential precision required by the existing (symbolic) algorithms for counting real zeros. The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time with an exponential number of processors.

cs.CC↗
arXiv2007

On the number of minima of a random polynomial

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain the bound O(exp(-beta n^2 + (n/2) log (d-1))) (beta is a positive constant independent on n and d) for the number of minima of such a polynomial. This proves that most normal random polynomials of fixed degree have only saddle points. Finally, we give a closed form expression for the number of maxima (resp. minima) of a random univariate polynomial, in terms of hypergeometric functions.

math.NA↗