Quantum annealing for systems of polynomial equations

We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations. Our first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial increase in size. In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size. A corollary of our first result is that the proof systems Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate each other in the presence of Boolean axioms.

The direct and inverse scattering problems in two-dimensional acoustics at a fixed frequency are considered. It is shown how the null-field approach can be modified so that the Q matrix (which in a straightforward manner gives the transition matrix) is obtained as a series instead of an integral. For an obstacle which is a perturbation of a circle this series form gives an approximate, very explicit, expression for the transition matrix. A few numerical examples are given to show the utility and limitations of this approximation. For the inverse problem, the series form of Q gives a system of nonlinear polynomial equations which are solved by the imbedding method. Some numerical examples show that quite accurate results are obtained by this method in cases where the system of equations can be kept small. The linear approximation of the system of polynomial equations yields a method that works surprisingly well and which is also promising for the more difficult three-dimensional and vector problems.

The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. This problem is NP-hard over any field. When the number of equations m is the same as the number of unknowns n the best known algorithms are exhaustive search for small fields, and a Gröbner base algorithm for large fields. Gröbner base algorithms have large exponential complexity and cannot solve in practice systems with n ≥ 15. Kipnis and Shamir [9] have recently introduced a new algorithm called “relinearization”. The exact complexity of this algorithm is not known, but for sufficiently overdefined systems it was expected to run in polynomial time.In this paper we analyze the theoretical and practical aspects of relinearization. We ran a large number of experiments for various values of n and m, and analysed which systems of equations were actually solvable. We show that many of the equations generated by relinearization are linearly dependent, and thus relinearization is less efficient that one could expect. We then develop an improved algorithm called XL which is both simpler and more powerful than relinearization. For all 0 < ε ≤ 1/2, and m ≥ εn 2, XL and relinearization are expected to run in polynomial time of approximately \( n^{\mathcal{O}(1/\sqrt \varepsilon )} \). Moreover, we provide strong evidence that relinearization and XL can solve randomly generated systems of polynomial equations in subexponential time when m exceeds n by a number that increases slowly with n.KeywordsQuadratic EquationExhaustive SearchAsymptotic ComplexityExact ComplexityMultivariate EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p ∈ P is seen by at least one guard g∈ G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃ ℝ, which has been studied earlier by other communities. The class ∃ ℝ consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP ⊆ ∃ ℝ. We prove that the art gallery problem is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃ ℝ. As an illustration of our techniques, we can show that for every compact semi-algebraic set S⊂ [0,1]2 there exists a polygon with rational coordinates that enforces one of the guards to be at any position p∈ S, in any optimal guarding. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem 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. In the ∃ ℝ-hardness proof for the art gallery problem we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it can be used to obtain ∃ ℝ-hardness proofs for other problems. In particular, ETR-INV has been very recently used to prove ∃ ℝ-hardness of other geometric problems.

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n 4 log n) + log w , the condition number k(M) satisfies k(M) ≤ w n 5/2 with probability at least 1-2w -1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length.

Systems of Polynomials for Detecting Orientable Matroids

**Read paper on the following link:** https://ifaamas.org/Proceedings/aamas2022/pdfs/p418.pdf **Abstract:** This paper presents a path-following combinatorial framework based on systems of polynomial equations to compute mixed Nash equilibria in $N$-person games. We provide the first detailed implementable description of Wilson's path-following method, which extends Lemke-Howson's algorithm to N-persons games and handles degenerate games as well. Our approach resembles, in some respects, support enumeration methods. We thus compare both approaches, theoretically and experimentally. Then, we show that the path-following approach allows to deal with a large family of succinctly expressed games: hypergraphical games, graphical games and polymatrix games. The described algorithms have been implemented in Python, making use of Sagemath libraries to solve systems of polynomial equations, allowing an experimental comparison of the different combinatorial approaches on a large variety of games.

Algebraic criteria for existence of scalar linear network codes to satisfy a set of connection requirements has been discussed extensively by Koetter and Meacutedard. Solving for a network code is now known to be equivalent to solving a system of polynomial equations obtained by assigning variables to edges in the line graph of the network and computing a suitable transfer function. An alternative formulation for arriving at an equivalent system of polynomial equations is given in this paper based on the decomposition of the original network into trees, which we call ldquoinformation flow treesrdquo. The basic idea is to exploit the graph structure and assign variables suitably. Interestingly, the information flow tree approach results in only linear and degree-2 equations that can be simplified considerably in directed acyclic networks as shown in prior work. In this article, we provide an alternative derivation of the information flow tree approach that results in two further extensions to networks with cycles. The first extension is to flow acyclic solutions on any directed cyclic network. The second extension is to cyclic networks where all strongly connected components are simple cycles. Here the degree of the equations we are left to solve is limited to 4.

Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two different ways: first, as a single, high-degree polynomial, and second as a collection of polynomials based on the complements of domination-critical graphs. We then provide a sufficient criterion for demonstrating that a particular ideal representation is already the universal Grobner bases of an ideal, and show that the second representation of the dominating set ideal in terms of domination-critical graphs is the universal Grobner basis for that ideal. We also present the first algebraic formulation of Vizing's conjecture, and discuss the theoretical and computational ramifications to this conjecture when using either of the two dominating set representations described above.

Solving equations—linear, algebraic, differential, difference, analytic, Diophantine …—is arguably the most central problem in mathematics. A case of this problem that can be efficiently tackled is that of linear systems of equations. What could be considered the level of difficulty immediately above that for linear systems, the case of quadratic, or more generally, polynomial equations, is substantially more complicated. Even for polynomials in one variable, classical results of Abel and Galois deprive us of any hope to actually compute their zeros. The best we can do is to approximate them (and a number of algorithms compute these approximations quite efficiently).For systems of multivariate polynomials we need to add complexity obstructions. The first that meets the eye is the possibly large number of solutions. A system of n quadratic equations in n variables has (generically) 2 n solutions in complex space \(\mathbb {C}^{n}\). But each polynomial in the system has \(\frac{1}{2}(n^{2}+3n+2)\) coefficients, and therefore the whole system is specified with Θ(n 3) coefficients. If we were to compute approximations for all its zeros, the size of the output would be exponential in the input size!A focal theme in this third part of the book is that of systems of polynomial equations and algorithms that approximate solutions of these systems. These algorithms have a “numeric” character, and it goes without saying that their analyses strongly rely on appropriate condition numbers. But the nature of these systems and their solutions suggests a view of their condition numbers within a more general framework than we encountered previously. The present chapter introduces this framework and provides some motivating (but also interesting per se) examples.

Recall that the reaction networks of deficiency zero can contain hundreds of species and hundreds of reactions. In such cases, the corresponding differential equations will be extraordinarily complex. When the kinetics is mass action, these will amount to a large and intricate system of coupled polynomial equations in perhaps 100 species concentrations and in which a large number of perhaps unknown parameters (rate constants) appear. Recall too that not much is known in general about systems of polynomial equations, even fairly small ones. Nevertheless, when these derive from deficiency zero networks, quite a lot can be said.

This paper presents a new analytical approach for a problem of coupling matrix (CM) synthesis. The approach is based on reformulation of well-known equations representing filter scattering parameters S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> and S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</inf> for a given coupling matrix. With the use of relation connecting inverse matrix its determinant and its adjugate matrix the system of equations is formed. The equations are reformulated in a way allowing direct comparison of polynomials coefficients for numerator and denominator of S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> and numerator of S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</inf> . This enables to form the system of polynomial equations, where unknowns are all entries of searched coupling matrix. In the proposed approach no conversion from S-parameters to Y-parameters is needed like in most of published techniques. SINGULAR software (GPL license) using Groebner basis for solution of such a system of polynomial equations was used. Two numerical experiments have been presented. First one CM synthesis for 4th order (one cross-coupling) filter. Second one is CM synthesis for 8th order (two cross-couplings) filter. Both experiments showed excellent results obtained with the proposed method. The method works for S-parameters of filters with arbitrary topologies.

Several recently proposed ciphers, for example Rijndael and Serpent, are built with layers of small S-boxes interconnected by linear key-dependent layers. Their security relies on the fact, that the classical methods of cryptanalysis (e.g. linear or differential attacks) are based on probabilistic characteristics, which makes their security grow exponentially with the number of rounds N r r. In this paper we study the security of such ciphers under an additional hypothesis: the S-box can be described by an overdefined system of algebraic equations (true with probability 1). We show that this is true for both Serpent (due to a small size of S-boxes) and Rijndael (due to unexpected algebraic properties). We study general methods known for solving overdefined systems of equations, such as XL from Eurocrypt’00, and show their inefficiency. Then we introduce a new method called XSL that uses the sparsity of the equations and their specific structure. The XSL attack uses only relations true with probability 1, and thus the security does not have to grow exponentially in the number of rounds. XSL has a parameter P, and from our estimations is seems that P should be a constant or grow very slowly with the number of rounds. The XSL attack would then be polynomial (or subexponential) in N r> , with a huge constant that is double-exponential in the size of the S-box. The exact complexity of such attacks is not known due to the redundant equations. Though the presented version of the XSL attack always gives always more than the exhaustive search for Rijndael, it seems to (marginally) break 256-bit Serpent. We suggest a new criterion for design of S-boxes in block ciphers: they should not be describable by a system of polynomial equations that is too small or too overdefined.

The elimination procedure as described by Williams has been coded in LISP and FORMAC and used in solving systems of polynomial equations. It is found that the method is very effective in the case of small systems, where it yields all solutions without the need for initial estimates. The method, by itself, appears inappropriate, however, in the solution of large systems of equations due to the explosive growth in the intermediate equations and the hazards which arise when the coefficients are truncated. A comparison is made with difficulties found in other problems in non-numerical mathematics such as symbolic integration and simplification.

In geodesy and geoinformatics, most problems are nonlinear in nature and often require the solution of systems of polynomial equations. Before 2002, solutions of such systems of polynomial equations, especially of higher degree remained a bottleneck, with iterative solutions being the preferred approach. With the entry of Groebner basis as algebraic solution to nonlinear systems of equations in geodesy and geoinformatics in the pioneering work “Grobner bases, multipolynomial resultants and the Gauss Jacobi combinatorial algorithms : adjustment of nonlinear GPS/LPS observations”, the playing field changed. Most of the hitherto unsolved nonlinear problems, e.g., coordinate transformation, global navigation satellite systems (GNSS)’s pseudoranges, resection-intersection in photogrammetry, and most recently, plane fitting in point clouds in laser scanning have been solved. A comprehensive overview of such applications are captured in the first and second editions of our book Algebraic Geodesy and Geoinformatics published by Springer. In the coming third edition, an updated summary of the newest techniques and methods of combination of Groenbner basis with symbolic as well as numeric methods will be treated. To quench the appetite of the reader, this presentation considers an illustrative example of a two-dimension coordinate transformation problem solved through the combination of symbolic regression and Groebner basis.