This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the associated spectral projections have entries in the ring. Building on this characterization, we develop an algorithm for deciding the unitary diagonalizability of matrices over regular local rings of algebraic varieties. A central ingredient of the algorithm is a decision procedure for determining whether a polynomial splits over a formal power series ring; we establish this using techniques from prime decomposition and the relative smoothness of integral closures in ramification theory.

This paper introduces a noncommutative version of the Nullstellensatz, motivated by the study of quantum nonlocal games. It has been proved that a two-answer nonlocal game with a perfect quantum strategy also admits a perfect classical strategy. We generalize this result to the infinite-dimensional case, showing that a two-answer game with a perfect commuting operator strategy also admits a perfect classical strategy. This result induces a special case of noncommutative Nullstellensatz.

Constraint-solving-based program invariant synthesis takes a parametric invariant template and encodes the (inductive) invariant conditions into constraints. The problem of characterizing the set of all valid parameter assignments is referred to as the strong invariant synthesis problem , while the problem of finding a concrete valid parameter assignment is called the weak invariant synthesis problem . For both problems, the challenge lies in solving or reducing the encoded constraints, which are generally non-convex and lack efficient solvers. In this article, we propose two novel algorithms for synthesizing invariants of polynomial programs using semidefinite programming (SDP): (1) The Cluster algorithm targets the strong invariant synthesis problem for polynomial invariant templates. Leveraging robust optimization techniques, it solves a series of SDP relaxations and yields a sequence of increasingly precise under-approximations of the set of valid parameter assignments. We prove the algorithm’s soundness, convergence, and weak completeness under a specific robustness assumption on templates. Moreover, the outputs can simplify the weak invariant synthesis problem. (2) The Mask algorithm addresses the weak invariant synthesis problem in scenarios where the aforementioned robustness assumption does not hold, rendering the Cluster algorithm ineffective. It identifies a specific subclass of invariant templates, termed masked templates, involving parameterized polynomial equalities and known inequalities. By applying variable substitution, the algorithm transforms constraints into an equivalent form amenable to SDP relaxations. Both algorithms have been implemented and demonstrated superior performance compared to state-of-the-art methods in our empirical evaluation.

Algebraic boundaries of convex semi-algebraic sets are closely related to polynomial optimization problems. Building upon Rainer Sinn’s work, we refine the stratification of iterated singular loci to a Whitney (a) stratification, which gives a list of candidates of varieties whose dual is an irreducible component of the algebraic boundary of the dual convex body. We also present an algorithm based on Teissier’s criterion to compute Whitney (a) stratifications, which employs conormal spaces and prime decomposition.

We study the unlabeled sensing problem that aims to solve a linear system of equations Ax = π(y) for an unknown permutation π. For a generic matrix A and a generic vector y, we construct a system of polynomial equations whose unique solution satisfies Aξ* = π(y). In particular, ξ* can be recovered by solving the rank-one moment matrix completion problem. We propose symbolic and numeric algorithms to compute the unique solution. Some numerical experiments are conducted to show the efficiency and robustness of the proposed algorithms.

We associate mirror games with the universal game algebra and use the *-representation to describe quantum commuting operator strategies. We provide an algebraic characterization of whether or not a mirror game has perfect commuting operator strategies. This new characterization uses a smaller algebra introduced by Paulsen and others for synchronous games and the noncommutative Nullstellensatz developed by Cimpric, Helton and collaborators. An algorithm based on noncommutative Gröbner basis computation and semidefinite programming is given for certifying that a given mirror game has no perfect commuting operator strategies.

Let f= (f1, ..., fs) be a sequence of polynomials in Q[X1,...,Xn] of maximal degree D and V⊂ Cn be the algebraic set defined by f and r be its dimension. The real radical re < f > associated to f is the largest ideal which defines the real trace of V . When V is smooth, we show that re < f >, has a finite set of generators with degrees bounded by V. Moreover, we present a probabilistic algorithm of complexity (snDn )O(1) to compute the minimal primes of re < f >. When V is not smooth, we give a probabilistic algorithm of complexity sO(1) (nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ Rn. Experiments are given to show the efficiency of our approaches.

We present an interactive probabilistic proof protocol that certifies in (log N)O(1) arithmetic and Boolean operations for the verifier the determinant, for example, of an N x N matrix over a field whose entries a(i,j) are given by a single (log NO(1)-depth arithmetic circuit, which contains (log NO(1) field constants and which is polynomial time uniform, for example, which has size (log NO(1). The prover can produce the interactive certificate within a (log NO(1) factor of the cost of computing the determinant. Our protocol is a version of the proofs for muggles protocol by Goldwasser, Kalai and Rothblum [STOC 2008, J. ACM 2015]. An application is the following: suppose in a system of k homogeneous polynomials of total degree ≤ d in the k variables y1,...,yk the coefficient of the term y1e1 ... ykek in the i-th polynomial is the (hypergeometric) value ((i+e1 + ... + ek)!)/((i!)(e1!)...(ek!)), where e! is the factorial of e. Then we have a probabilistic protocol that certifies (projective) solvability or inconsistency of such a system in (k log(d))O(1) bit complexity for the verifier, that is, in polynomial time in the number of variables k and the logarithm of the total degree, log(d).

Ankur Moitra in his paper at STOC 2015 has given an in-depth analysis of how oversampling improves the conditioning of the arising Prony systems for sparse interpolation and signal recovery from numeric data. Moitra assumes that oversampling is done for a number of samples beyond the actual sparsity of the polynomial/signal. We give an algorithm that can be used to compute the sparsity and estimate the minimal number of samples needed in numerical sparse interpolation. The early termination strategy of polynomial interpolation has been incorporated in the algorithm: by oversampling at a small number of extra sample points we can diagnose that the sparsity has not been reached.

We consider the problem of optimizing a parametric linear function over a non-compact real trace of an algebraic set. Our goal is to compute a representing polynomial which defines a hypersurface containing the graph of the optimal value function. Rostalski and Sturmfels showed that when the algebraic set is irreducible and smooth with a compact real trace, then the least degree representing polynomial is given by the defining polynomial of the irreducible hypersurface dual to the projective closure of the algebraic set.

In this talk, we will introduce some recent progress in dealing with optimization problems over noncompact semialgebraic sets. We will start with the problem of optimizing a parametric linear function over a noncompact real algebraic variety. Then we will introduce how to compute the semidefinite representation or approximation of the convex hull of a noncompact semialgebraic set. Finally, we will show how to characterize the lifts of noncompact convex sets by the cone factorizations of properly defined slack operators.

In the tutorial, we will introduce two kinds of problems for which validated results are computed via hybrid symbolic-numeric algorithms. These hybrid algorithms follow the basic principle pointed out by Siegfried M. Rump in [1] for computing validated results: First, a pure floating point algorithm is used to compute an approximate solution of good quality for a given problem. Second, a verification step using exact rational arithmetic or interval arithmetic is appended. If this step is successful, then certified lower bounds or verified error bounds are computed for the previously computed approximation.

Consider a (D x D) symmetric matrix A whose entries are linear forms in Q[X1, ..., Xk] with coefficients of bit size ≤ τ. We provide an algorithm which decides the existence of rational solutions to the linear matrix inequality A ≥ 0 and outputs such a rational solution if it exists. This problem is of first importance: it can be used to compute algebraic certificates of positivity for multivariate polynomials. Our algorithm runs within (k≤)O(1)2O(min(k, D)D2)DO(D2) bit operations; the bit size of the output solution is dominated by τO(1)2O(\min(k, D)D2). These results are obtained by designing algorithmic variants of constructions introduced by Klep and Schweighofer. This leads to the best complexity bounds for deciding the existence of sums of squares with rational coefficients of a given polynomial. We have implemented the algorithm; it has been able to tackle Scheiderer's example of a multivariate polynomial that is a sum of squares over the reals but not over the rationals; providing the first computer validation of this counter-example to Sturmfels' conjecture.

In this paper, we propose two algorithms for verifying the existence of real solutions of positive-dimensional polynomial systems. The first one is based on the critical point method and the homotopy continuation method. It targets for verifying the existence of real roots on each connected component of an algebraic variety V ∩ Rn defined by polynomial equations. The second one is based on the low-rank moment matrix completion method and aims for verifying the existence of at least one real roots on V ∩ Rn. Combined both algorithms with the verification algorithms for zero-dimensional polynomial systems, we are able to find verified real solutions of positive-dimensional polynomial systems very efficiently for a large set of examples.

We deploy numerical semidefinite programming and conversion to exact rational inequalities to certify that for a positive semidefinite input polynomial or rational function, any representation as a fraction of sums-of-squares of polynomials with real coefficients must contain polynomials in the denominator of degree no less than a given input lower bound. By Artin's solution to Hilbert's 17th problems, such representations always exist for some denominator degree. Our certificates of infeasibility are based on the generalization of Farkas's Lemma to semidefinite programming.

In this paper, we propose a new algorithm for computing real roots of polynomial equations or a subset of real roots in a given semi-algebraic set described by additional polynomial inequalities. The algorithm is based on using modified fixed point continuation method for solving Lasserre's hierarchy of moment relaxations. We establish convergence properties for our algorithm. For a large-scale polynomial system with only few real solutions in a given area, we can extract them quickly. Moreover, for a polynomial system with an infinite number of real solutions, our algorithm can also be used to find some isolated real solutions or real solutions on the manifolds.

The problem of computing a representation for a real polynomial as a sum of minimum number of squares of polynomials can be casted as finding a symmetric positive semidefinite real matrix of minimum rank subject to linear equality constraints. In this paper, we propose algorithms for solving the minimum-rank Gram matrix completion problem, and show the convergence of these algorithms. Our methods are based on the fixed point continuation method. We also use the Barzilai-Borwein technique and a specific linear combination of two previous iterates to accelerate the convergence of modified fixed point continuation algorithms. We demonstrate the effectiveness of our algorithms for computing approximate and exact rational sum of squares decompositions of polynomials with rational coefficients.

Let X = [X1, ..., Xn] and f ∈ R[X]. We consider the problem of computing the global infimum of f when f is bounded below. For A ∈ GLn(C), we denote by fA the polynomial f(A X). Fix a number M ∈ R greater than infx∈Rn f(x). We prove that there exists a Zariski-closed subset A [equation] GLn(C) such that for all A ∈ GLn(Q) \ A, we have fA ≥ 0 on Rn if and only if for all ε > 0, there exist sums of squares of polynomials s and t in R[X] and polynomials [Equation]. Hence we can formulate the original optimization problems as semidefinite programs which can be solved efficiently in Matlab. Some numerical experiments are given. We also discuss how to exploit the sparsity of SDP problems to overcome the ill-conditionedness of SDP problems when the infimum is not attained.

We give a stability criterion for real polynomial inequalities with floating point or inexact scalars by estimating from below or computing the radius of semidefiniteness. That radius is the maximum deformation of the polynomial coefficient vector measured in a weighted Euclidean vector norm within which the inequality remains true. A large radius means that the inequalities may be considered numerically valid.

The problem of blind image deconvolution can be solved by computing approximate greatest common divisors (GCD) of polynomials. The bivariate polynomials corresponding to the z-transforms of several blurred images have an approximate GCD corresponding to the z-transform of the original image. Since blurring functions as cofactors have very low degree in general, this GCD will be of high degree. On the other hand, if we only have one blurred image and want to identify the original scene, the blurred image can be partitioned such that each part completely contains the blurring function, hence the blurring function becomes the GCD which is of low degree. Therefore, we design a specialized algorithm for computing GCDs of polynomials to recover true images in two different cases. The new algorithm is based on the fast GCD algorithm for univariate polynomials and the Fast Fourier Transform (FFT) algorithm. The complexity of our specialized algorithm for identifying both the true image and the blurring functions from blurred images of size n x n is O(n2 log(n)) in the case of blurring functions of very low degree. The algorithm has been implemented in Maple and can extract true images of hundreds by hundreds pixel images from blurred images in a few seconds.

We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several well-known factorization problems in hybrid symbolic-numeric computation. The idea is to transform a numerical sum-of-squares (SOS) representation of a positive polynomial into an exact rational identity. Our algorithms successfully certify accurate rational lower bounds near the irrational global optima for benchmark approximate polynomial greatest common divisors and multivariate polynomial irreducibility radii from the literature, and factor coefficient bounds in the setting of a model problem by Rump (up to n = 14, factor degree = 13.

We present a method based on symbolic-numeric reduction to geometric involutive form to compute the primary component and the differential operators f solution of a polynomial ideal. The singular solution can be exact or approximate. If the singular solution is known with limited accuracy, then we propose a new method to refine it to high accuracy.

We consider the problem of computing minimal real or complex deformations to the coefficients in a list of relatively prime real or complex multivariate polynomials such that the deformed polynomials have a greatest common divisor (GCD) of at least a given degree k. In addition, we restrict the deformed coefficients by a given set of linear constraints, thus introducing the linearly constrained approximate GCD problem. We present an algorithm based on a version of the structured total least norm (STLN) method and demonstrate on a diverse set of benchmark polynomials that the algorithm in practice computes globally minimal approximations. As an application of the linearly constrained approximate GCD problem we present an STLN-based method that computes a real or complex polynomial the nearest real or complex polynomial that has a root of multiplicity at least k. We demonstrate that the algorithm in practice computes on the benchmark polynomials given in the literature the known globally optimal nearest singular polynomials. Our algorithms can handle, via randomized preconditioning, the difficult case when the nearest solution to a list of real input polynomials actually has non-real complex coefficients.

Several standard problems in symbolic computation, such as greatest common divisor and factorization of polynomials, sparse interpolation, or computing solutions to overdetermined systems of polynomial equations have non-trivial solutions only if the input coefficients satisfy certain algebraic constraints. Errors in the coefficients due to floating point round-off or through phsical measurement thus render the exact symbolic algorithms unusable. By symbolic-numeric methods one computes minimal deformations of the coefficients that yield non-trivial results. We will present hybrid algorithms and benchmark computations based on Gauss-Newton optimization, singular value decomposition(SVD) and structure-preserving total least squares (STLS) fitting for several of the above problems.A significant body of results to solve those "approximate computer algebra" problems has been discovered in the past 10 years. In the Computer Algebra Handbook the section on "Hybrid Methods" concludes as follows [2]: "The challenge of hybrid symbolic-numeric algorithms is to explore the effects of imprecision, discontinuity, and algorithmic complexity by applying mathematical optimization, perturbation theory, and inexact arithmetic and other tools in order to solve mathematical problems that today are not solvable by numerical or symbolic methods alone." The focus of our tutorial is on how to formulate several approximate symbolic computation problems as numerical problems in linear algebra and optimization and on software that realizes their solutions.Approximate Greatest Common Divisors [3]. Our paper at this conference presents a solution to the approximate GCD problem for several multivariate polynomials with real or complex coefficients. In addition, the coefficients of the minimally deformed input coefficients can be linearly constrained. In our tutorial we will give a precise definition of the approximate polynomial GCD problem and we will present techniques based on parametric optimization (slow) and STLS or Gauss/Newton iteration (fast) for its numerical solution. The fast methods can compute globally optimal solutions, but they cannot verify global optimality. We show how to apply the constrained approximate GCD problem to computing the nearest singular polynomial with a root of multiplicity at least k≥2.Approximate Factorization of Multivariate Polynomials [1]. Our solution and implementation of the approximate factorization problem follows our approach for the approximate GCD problem. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use SVD or STLS and Gauss/Newton optimization to numerically compute the approximate multivariate factors.Solutions of Zero-dimensional Polynomial Systems [4]. We translate a system of polynomials into a system of linear partial differential equations (PDEs) with constant coefficients. The PDEs are brought to an involutive form by symbolic prolongations and numeric projections via SVD. The solutions of the polynomial system are obtained by solving an eigen-problem constructed from the null spaces of the involutive system and its geometric projections.

The input to our algorithm is a multivariate polynomial, whose complex rational coefficients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coefficients by a small quantitity such that the resulting polynomial factors over C. Ideally, one would like to minimize the perturbation in some selected distance measure, but no efficient algorithm for that is known. We give a numerical multivariate greatest common divisor algorithm and use it on a numerical variant of algorithms by W. M. Ruppert and S. Gao. Our numerical factorizer makes repeated use of singular value decompositions. We demonstrate on a significant body of experimental data that our algorithm is practical and can find factorizable polynomials within a distance that is about the same in relative magnitude as the input error, even when the relative error in the input is substantial (10-3).

Camera pose estimation is the problem of determining the position and orientation of an internally calibrated camera from known 3D reference points and their images. We briefly survey several existing methods for pose estimation, then introduce our new complete linear method, which is based on a symbolic-numeric method from the geometric (Jet) theory of partial differential equations. The method is stable and robust. In particular, it can deal with the points near critical configurations. Numerical experiments are given to show the performance of the new method.

We treat pseudo-factorization of a multivariate polynomial p over C as an overdetermined bilinear system for the coefficients of the factors. If the specified data (coefficients of p) are sufficiently close to the manifold of consistent data we project onto it by solving a well-chosen subsystem (otherwise we quit). On the manifold, we reduce the backward error by a linearized minimization procedure. The resulting algorithm appears to extend the range of treatable cases - in terms of number of variables and total degree - considerably.