Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

The Farthest Spatial Skyline Queries

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [ 7 ], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [ 25 ] and Kedlaya & Umans [ 16 ] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most \(d^{o(1)}\) where the degree of the input polynomial in every variable is less than d . They also stated the question of designing fast algorithms for the large variable case (i.e., \(n \notin d^{o(1)}\) ) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field \(\mathbb {F}_{q}\) of characteristic p , which evaluates an n -variate polynomial of degree less than d in each variable on N inputs in time \(\begin{equation*} \left((N + d^n)^{1 + o(1)}\text{poly}(\log q, d, n, p)\right), \end{equation*}\) provided that p is at most d o (1) , and q is at most (exp (exp (exp (...(exp ( d ))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n ∉ d o (1) ), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: — We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [ 21 ] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. — We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant’s program [ 26 ]. More precisely, for every fixed prime p , we show that for every constant ɛ > 0, and large enough n , the rank of any \(n \times n\) Vandermonde matrix V over the field \(\mathbb {F}_{p^a}\) can be reduced to ( n /exp (Ω (poly(ɛ)log 0.53 n ))) by changing at most n Θ (ɛ) entries in every row of V , provided a ≤ poly(log n ). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [ 9 ].

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground elds to arbitrary non-archimedean valued elds. To achieve this, we develop a theory of toric schemes over valuation rings of rank 1. As a basic tool, we use techniques from non-archimedean analysis. MSC2010: 14T05, 14M25, 32P05 X) of polyhedra in R n . This process is called tropicaliza- tion and it can be used to transform a problem from algebraic geometry into a corresponding problem in convex geometry which is usually easier. If the toric co- ordinates are well suited to the problem, it is sometimes possible to use a solution of the convex problem to solve the original algebraic problem. Another strategy is to vary the ambient torus to compensate the loss of information due to the tropi- calization process. Tropicalization originates from a paper of Bergman (Berg) on logarithmic limit sets. The convex structure of the tropical variety Trop(X) was worked out by Bieri{Groves (BG) with applications to geometric group theory in mind. Sturmfels (Stu) pointed out that Trop(X) is a subcomplex of the Grobner complex. In fact, the polyhedral complex Trop(X) has some natural weights satisfying a balancing condition which appears rst in Speyer's thesis (Spe). This relies on the description

We present a new approach to the problem of computing the zeta function of a hypersurface over a finite field. For a hypersurface defined by a polynomial of degree d in n variables over the field of q elements, one desires an algorithm whose running time is a polynomial function of dn log(q). (Here we assume d ⩾ 2, for otherwise the problem is easy.) The case n = 1 is related to univariate polynomial factorisation and is comparatively straightforward. When n = 2 one is counting points on curves, and the method of Schoof and Pila yields a complexity of log ⁡ ( q ) C d , where the function Cd depends exponentially on d. For arbitrary n, the theorem of the author and Wan gives a complexity which is a polynomial function of (pdn log(q))n, where p is the characteristic of the field. A complexity estimate of this form can also be achieved for smooth hypersurfaces using the method of Kedlaya, although this has only been worked out in full for curves. The new approach we present should yield a complexity which is a small polynomial function of pdn log(q). In this paper, we work this out in full for Artin–Schreier hypersurfaces defined by equations of the form Zp − Z = f, where the polynomial f has a diagonal leading form. The method utilises a relative p-adic cohomology theory for families of hypersurfaces, due in essence to Dwork. As a corollary of our main theorem, we obtain the following curious result. Let f be a multivariate polynomial with integer coefficients whose leading form is diagonal. There exists an explicit deterministic algorithm which takes as input a prime p, outputs the number of solutions to the congruence equation f = 0 op, and runs in {\mathcal O}(p^{2 + \varepsilon}) bit operations, for any ε > 0. This improves upon the elementary estimate of {\mathcal O}(p^{n-1 + \varepsilon}) bit operations, where n is the number of variables, which can be achieved using Berlekamp's root counting algorithm. 2000 Mathematics Subject Classification 11Y99, 11M38, 11T99.

Many approximation algorithms have been presented in the last decades for ${\cal NP}$-hard search problems. The focus of this paper is on cryptographic applications, where it is desirable to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms—efficient approximation algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, Feigenbaum et al. [ACM Trans. Algorithms, 2 (2006), pp. 435–472] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to relations is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms; applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. We argue that our privacy requirement is natural and quite minimal. We show that, even under this minimal definition of privacy, for well-studied problems such as vertex cover and max exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similarly to Halevi et al. [in Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 550–559], we define a relaxed notion of approximation algorithms that leak (a little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for max exact 3SAT that leak a little information.

This paper gives a tutorial presentation of the canonical variate analysis (CVA) method of subspace system identification for linear time- invariant (LTI), linear parameter-varying (LPV), and nonlinear parameter-varying (NLPV) systems. The paper takes a first principles statistical approach to rank determination of deterministic vectors usinig a singular value decomposition (SVD), followed by a statistical multivariate CVA as rank constrained regression. This is generalized to LTI dynamic systems, and extended directly to LPV and NLPV. The computational structure and problem size is very similar to LTI subspace methods except that the matrix row dimension (number of lagged variables of the past) is multiplied by the effective number of parameter-varying functions. This is in contrast with the exponential explosion in the number of variables using current subspace methods for LPV systems. Compared with current methods, initial results indicate much less computation, maximum likelihood accuracy, and better numerical stability. The method applies to systems with feedback, and can automatically remove a number of redundancies in the nonlinear models producing near minimal state orders and polynomial degrees by hypothesis testing. There is detailed discussion of the methods and structure of the computational modules.

Written to honor the 80th birthday of William Fulton, the articles collected in this volume (the first of a pair) present substantial contributions to algebraic geometry and related fields, with an emphasis on combinatorial algebraic geometry and intersection theory. Featured topics include commutative algebra, moduli spaces, quantum cohomology, representation theory, Schubert calculus, and toric and tropical geometry. The range of these contributions is a testament to the breadth and depth of Fulton's mathematical influence. The authors are all internationally recognized experts, and include well-established researchers as well as rising stars of a new generation of mathematicians. The text aims to stimulate progress and provide inspiration to graduate students and researchers in the field.

We characterize the optimal solution to the LP relaxation of the standard formulation for the minimum satisfiability problem. Based on the characterization, we give a \(O(nm^2)\) combinatorial algorithm to solve the fractional version of the minimum satisfiability problem optimally where n(m) is the number of variables (clauses). As a by-product, we obtain a \(2(1-1/2^k)\) approximation algorithm for the minimum satisfiability problem where k is the maximum number of literals in any clause. We also give a simple linear time 2 approximation algorithm.

It is known that random $k$-Sat instances with at least $(2^k \cdot \ln 2)\cdot n$ random clauses are unsatisfiable with high probability. This result is simply obtained by showing that the expected number of satisfying assignments tends to $0$ when the number of variables $n$ tends to infinity. This proof does not directly provide us with an efficient algorithm certifying the unsatisfiability of a given random formula. Concerning efficient algorithms, it is essentially known that random formulas with $n^\varepsilon \cdot n^{k/2}$ clauses with $k$ literals can be efficiently certified as unsatisfiable. The present paper is the result of trying to lower this bound. We obtain better bounds for some specialized satisfiability problems. These results are based on discrepancy investigations for hypergraphs. Further, we show that random formulas with a linear number of clauses can be efficiently certified as unsatisfiable in the Not-All-Equal-$3$-Sat sense. A similar result holds for the non-$3$-colourability of random graphs with a linear number of edges. We obtain these results by direct application of approximation algorithms.

We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.

Non-Hermitian systems have been widely explored in platforms ranging from photonics to electric circuits. A defining feature of non-Hermitian systems is exceptional points (EPs), where both eigenvalues and eigenvectors coalesce. Tropical geometry is an emerging field of mathematics at the interface between algebraic geometry and polyhedral geometry, with diverse applications to science. Here, we introduce and develop a unified tropical geometric framework to characterize different facets of non-Hermitian systems. We illustrate the versatility of our approach using several examples and demonstrate that it can be used to select from a spectrum of higher-order EPs in gain and loss models, predict the skin effect in the non-Hermitian Su-Schrieffer-Heeger model, and extract universal properties in the presence of disorder in the Hatano-Nelson model. Our work puts forth a framework for studying non-Hermitian physics and unveils a connection of tropical geometry to this field.

In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my talks at the 2015 AMS symposium in algebraic geometry and at AGNES 2016, is to present some of the results in this area.

We consider an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0-1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function or that of a reformulated objective function.

We present a new family of sums of squares (SOS) relaxations to cones of positive polynomials. The SOS relaxations employed in the literature are cones of polynomials which can be represented as ratios, with an SOS as numerator and a fixed positive polynomial as denominator. We employ nonlinear transformations of the arguments instead. A fixed cone of positive polynomials, considered as a subset in an abstract coefficient space, corresponds to an infinite, partially ordered set of concrete cones of positive polynomials of different degrees and in a different number of variables. To each such concrete cone corresponds its own SOS cone, leading to a hierarchy of increasingly tighter SOS relaxations for the abstract cone.

The Weil bound for character sums is a deep result in Algebraic Geometry with many applications both in mathematics and in the theoretical computer science. The Weil bound states that for any polynomial $f(x)$ over a finite field $\mathbb{F}$ and any additive character $\chi:\mathbb{F} \to \mathbb{C}$, either $\chi(f(x))$ is a constant function or it is distributed close to uniform. The Weil bound is quite effective as long as $\deg(f) \ll \sqrt{|\mathbb{F}|}$, but it breaks down when the degree of $f$ exceeds $\sqrt{|\mathbb{F}|}$. As the Weil bound plays a central role in many areas, finding extensions for polynomials of larger degree is an important problem with many possible applications. In this work we develop such an extension over finite fields $\mathbb{F}_{p^n}$ of small characteristic: we prove that if $f(x)=g(x)+h(x)$ where $\deg(g) \ll \sqrt{|\mathbb{F}|}$ and $h(x)$ is a sparse polynomial of arbitrary degree but bounded weight degree, then the same conclusion of the classical Weil bound still holds: either $\chi(f(x))$ is constant or its distribution is close to uniform. In particular, this shows that the sub code of Reed-Muller codes of degree $\omega(1)$ generated by traces of sparse polynomials is a code with near optimal distance, while Reed-Muller of such a degree has no distance (i.e. $o(1)$ distance), this is one of the few examples where one can prove that sparse polynomials behave differently from non-sparse polynomials of the same degree. As an application we prove new general results for affine invariant codes. We prove that any affine-invariant subspace of quasi-polynomial size is (1) indeed a code (i.e. has good distance) and (2) is locally testable. Previous results for general affine invariant codes were known only for codes of polynomial size, and of length $2^n$ where $n$ needed to be a prime. Thus, our techniques are the first to extend to general families of such codes of super-polynomial size, where we also remove the requirement from $n$ to be a prime. The proof is based on two main ingredients: the extension of the Weil bound for character sums, and a new Fourier-analytic approach for estimating the weight distribution of general codes with large dual distance, which may be of independent interest.