We analyze the decomposition problem of multivariate polynomial-exponential functions from truncated series and present new algorithms to compute their decomposition. Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial-exponential functions. They are also the solutions of systems of partial differential equations with constant coefficients. We relate their representation to the inverse system of the roots of the characteristic variety. Using the properties of Hankel operators, we establish a correspondence between polynomial exponential series and Artinian Gorenstein algebras. We generalize Kronecker theorem to the multivariate case, by showing that the symbol of a Hankel operator of finite rank is a polynomial-exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol. A generalization of Prony's approach to multivariate decomposition problems is presented , exploiting eigenvector methods for solving polynomial equations. We show how to compute the frequencies and weights of a minimal polynomial-exponential decomposition , using the first coefficients of the series. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Carath 'e odory-Fej 'e r decomposition of multivariate Hankel matrices. A new algorithm is given to compute a basis of the Artinian Gorenstein algebra, based on a Gram-Schmidt orthogonalization process and to decompose polynomial-exponential series. A general framework for the applications of this approach is described and illustrated in different problems. We provide Kronecker-type theorems for convolution operators, showing that a convolution operator (or a cross-correlation operator) is of finite rank, if and only if, its symbol is a polynomial-exponential function, and we relate its rank to the decomposition of its symbol. We also present Kronecker-type theorems for the reconstruction of measures as weighted sums of Dirac measures from moments and for the decomposition of polynomial-exponential functions from values. Finally, we describe an application of this method for the sparse interpolation of polylog functions from values.
This type of approaches is also used in sparse interpolation of black box polynomials. In the methods developed in @cite_24 , @cite_32 , further improved in @cite_20 , the sparse polynomial is evaluated at points of the form @math where @math are prime numbers or primitive roots of unity of co-prime order. The sparse decomposition of the black box polynomial is computed from its values by a univariate Prony-like method.
@cite_24 @cite_32 @cite_20
1599026900 1977375620
We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or Tiwari sparse interpolation algorithm and the classical Prony's method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony's method. We analyse the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications. Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm sequences, Continued Fractions and Descartes' rule of sign. The methods under consideration were developed at the GALAAD group at INRIA,the VEGAS group at LORIA and the MPI Saarbrucken. We compared their sensitivity with respect to various aspects such as degree, bitsize or root separation of the input polynomials. Our datasets consist of 5,000 polynomials from many different settings, which have maximum coefficient bitsize up to bits 8,000, and the total running time of the experiments was about 50 hours. Thereby, all implementations of the theoretically exact methods always provided correct results throughout this extensive study. For each scenario we identify the currently most adequate method, and we point to weaknesses in each approach, which should lead to further improvements. Our results indicate that there is no "best method" overall, but one can say that for most instances the solvers based on Continued Fractions are among the best methods. To the best of our knowledge, this is the largest number of tests for univariate real solving up to date. In this paper we study polynomial identity testing of sums of @math read-once algebraic branching programs ( @math -RO-ABPs), generalizing the work in (Shpilka and Volkovich 2008,2009), who considered sums of @math read-once formulas ( @math -RO-formulas). We show that @math -RO-ABPs are strictly more powerful than @math -RO-formulas, for any @math , where @math is the number of variables. We obtain the following results: 1) Given free access to the RO-ABPs in the sum, we get a deterministic algorithm that runs in time @math , where @math bounds the size of any largest RO-ABP given on the input. This implies we have a deterministic polynomial time algorithm for testing whether the sum of a constant number of RO-ABPs computes the zero polynomial. 2) Given black-box access to the RO-ABPs computing the individual polynomials in the sum, we get a deterministic algorithm that runs in time @math . 3) Finally, given only black-box access to the polynomial computed by the sum of the @math RO-ABPs, we obtain an @math time deterministic algorithm. Let @math be an arbitrary polynomial of degree @math with @math non-zero integer coefficients of absolute value less than @math . In this paper, we answer the open question whether the real roots of @math can be computed with a number of arithmetic operations over the rational numbers that is polynomial in the input size of the sparse representation of @math . More precisely, we give a deterministic, complete, and certified algorithm that determines isolating intervals for all real roots of @math with @math many exact arithmetic operations over the rational numbers. When using approximate but certified arithmetic, the bit complexity of our algorithm is bounded by @math , where @math means that we ignore logarithmic. Hence, for sufficiently sparse polynomials (i.e. @math for a positive constant @math ), the bit complexity is @math . We also prove that the latter bound is optimal up to logarithmic factors.