On the total order of reducibility of a pencil of algebraic plane curves

In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d^2-1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account the Newton's polygon of the pencil

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

Finding polynomial loop invariants for probabilistic programs

Quantitative loop invariants are an essential element in the verification of probabilistic programs. Recently, multivariate Lagrange interpolation has been applied to synthesizing polynomial invariants. In this paper, we propose an alternative approach. First, we fix a polynomial template as a candidate of a loop invariant. Using Stengle's Positivstellensatz and a transformation to a sum-of-squares problem, we find sufficient conditions on the coefficients. Then, we solve a semidefinite programming feasibility problem to synthesize the loop invariants. If the semidefinite program is unfeasible, we backtrack after increasing the degree of the template. Our approach is semi-complete in the sense that it will always lead us to a feasible solution if one exists and numerical errors are small. Experimental results show the efficiency of our approach.Comment: accompanies an ATVA 2017 submissio

On the complexity of computing real radicals of polynomial systems

International audienceLet 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 associated to f is the largest ideal which defines the real trace of V . When V is smooth, we show that re , 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 . 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

On the Generation of Positivstellensatz Witnesses in Degenerate Cases

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.Comment: To appear in ITP 201

A Practical Approach to the Secure Computation of the Moore-Penrose Pseudoinverse over the Rationals

Solving linear systems of equations is a universal problem. In the context of secure multiparty computation (MPC), a method to solve such systems, especially for the case in which the rank of the system is unknown and should remain private, is an important building block. We devise an efficient and data-oblivious algorithm (meaning that the algorithm\u27s execution time and branching behavior are independent of all secrets) for solving a bounded integral linear system of unknown rank over the rational numbers via the Moore-Penrose pseudoinverse, using finite-field arithmetic. I.e., we compute the Moore-Penrose inverse over a finite field of sufficiently large order, so that we can recover the rational solution from the solution over the finite field. While we have designed the algorithm with an MPC context in mind, it could be valuable also in other contexts where data-obliviousness is required, like secure enclaves in CPUs. Previous work by Cramer, Kiltz and Padró (CRYPTO 2007) proposes a constant-rounds protocol for computing the Moore-Penrose pseudoinverse over a finite field. The asymptotic complexity (counted as the number of secure multiplications) of their solution is $O(m^4 + n^2 m)$, where $m$ and $n$, $m\leq n$, are the dimensions of the linear system. To reduce the number of secure multiplications, we sacrifice the constant-rounds property and propose a protocol for computing the Moore-Penrose pseudoinverse over the rational numbers in a linear number of rounds, requiring only $O(m^2n)$ secure multiplications. To obtain the common denominator of the pseudoinverse, required for constructing an integer-representation of the pseudoinverse, we generalize a result by Ben-Israel for computing the squared volume of a matrix. Also, we show how to precondition a symmetric matrix to achieve generic rank profile while preserving symmetry and being able to remove the preconditioner after it has served its purpose. These results may be of independent interest

Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Groebner Bases

We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the THEOREMA system; some code fragments and sample computations are included