depth of proofs depth of cut formulas and complexity of cut formulas abstract the relation between proofs with cuts and proofs without cuts is discussed in this article the complexity of cut formulas is redefined to better reflect the structure of cut formulas which is important to cut elimination a cut elimination strategy based on this definition and an upper bound of cut elimination are given further it is explained that cut formulas with depth larger than the depth of the proof can be shortened without increasing the depth of the proof and the complexity of the cut formulas a new upper bound based on this fact and the previous upper bound is given

approximate gcd and its application to ill conditioned algebraic equations abstract we describe two algorithms of approximate gcd greatest common divisor for polynomials with coefficients of floating point numbers one is for univariate polynomials and the other for multivariate polynomials the algorithms are careful extensions of the conventional euclidean algorithm and they are applied to solving ill conditioned algebraic equations ill conditioned univariate algebraic equations have multiple or close roots and we can ill conditioned system of algebraic equations which have an approprimate common divisor conventional numeric root finding methods such as newton s method give no satisfactory result for such ill conditioned systems by using an approximate gcd algorithm for multivariate polynomials the system is transformed into a well conditioned problems by a combination of algebraic and numeric methods we call such algorithms hybrid algorithms this paper shows the importance and possible fruitfulness of the hybrid algorithm

an improved non linear method for the computation of a structured low rank approximation of the sylvester resultant matrix this paper reports on improvements to recent work on the computation of a structured low rank approximation of the sylvester resultant matrix s f g of two inexact polynomials f f y and g g y specifically it has been shown in previous work that these polynomials must be processed before a structured low rank approximation of s f g is computed the existing algorithm may still however yield a structured low rank approximation of s f g but not a structured low rank approximation of s g f which is unsatisfactory moreover a structured low rank approximation of s f g must be equal to apart from permutations of its columns a structured low rank approximation of s g f but the existing algorithm does not guarantee the satisfaction of this condition this paper addresses these issues by modifying the existing algorithm such that these deficiencies are overcome examples that illustrate these improvements are shown

blind image deconvolution using a robust gcd approach in this correspondence a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions in z domain the desired image can be regarded as the greatest common polynomial divisor among the distorted versions with the assumption that the distortion filters are finite impulse response fir and relatively coprime in the absence of noise this becomes a problem of taking the greatest common divisor gcd of two or more two dimensional 2 d polynomials exact gcd is not desirable because even extremely small variations due to quantization error or additive noise can destroy the integrity of the polynomial system and lead to a trivial solution our approach to this blind deconvolution approximation problem introduces a new robust interpolative 2 d gcd method based on a one dimensional 1 d sylvester type gcd algorithm experimental results with both synthetically blurred images and real motion blurred pictures show that it is computationally efficient and moderately noise robust

an elementary proof of barnett s theorem about the greatest common divisor of several univarlate polynomials abstract this articule provides a new proof of barnett s theorem giving the degree of the greatest common divisor of several univariate polynomials with coefficients in a field in terms of the rank of a well defined matrix the new proof is elementary and self contained no use of jordan form or invariant factors and it is based on some easy to state properties of subresultants moreover this proof allows one to generalize barnett s results to the case when the considered polynomials have their coefficients in an integral domain

common factor detection and estimation the paper introduces a noniterative maximum likelihood ml based approach for the detection and estimation of the greatest common divisor of two given polynomials the coefficients of which are assumed to have a normal distribution with a consistently estimable covariance matrix it then goes on to show how this approach can be used to derive a noniterative ml method for estimating the orders and parameters of difference equation models with autoregressive residuals numerical examples are included to illustrate the performance of the proposed methodology

a gersgorin type lower bound for the smallest singular value abstract for an n by m matrix a aij n m we show that the smallest singular value of a is bounded below by min 1 i n a ii 1 2 j 1 j i n a ij j 1 j i n a ji this improves upon two known lower bounds without increasing information requirements or complexity of calculation

a fast recursive orthogonalization scheme for the macaulay matrix in this article we present a fast recursive orthogonalization scheme for two important subspaces of the macaulay matrix its row space and null space it requires a graded monomial ordering and exploits the resulting structure of the macaulay matrix induced by this graded ordering the resulting orthogonal basis for the row space will retain a similar structure as the macaulay matrix and is as a consequence sparse the computed orthogonal basis for the null space is dense but typically has smaller dimensions two alternative implementations for the recursive orthogonalization scheme are presented one using the singular value decomposition and another using a sparse rank revealing multifrontal qr decomposition numerical experiments show the eectiveness of the proposed recursive orthogonalization scheme in both running time and required memory compared to a standard orthogonalization the sparse multifrontal qr implementation is superior in both total run time and required memory at the cost of being slightly less reliable for determining the numerical rank

qr factoring to compute the gcd of univariate approximate polynomials we present a stable and practical algorithm that uses qr factors of the sylvester matrix to compute the greatest common divisor gcd of univariate approximate polynomials over spl ropf x or spl copf x an approximate polynomial is a polynomial with coefficients that are not known with certainty the algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle by splitting and reversal if necessary the algorithm has been tested on thousands of examples including pairs of polynomials of up to degree 1000 and is now distributed as the program qrgcd in the snap package of maple 9

approximate least common multiple of several polynomials using the eres division algorithm in this paper a numerical method for the computation of the approximate least common multiple alcm of a set of several univariate real polynomials is presented the most important characteristic of the proposed method is that it avoids root finding procedures and computations involving the greatest common divisor gcd conversely it is based on the algebraic construction of a special matrix which contains key data from the original set of polynomials and leads to the formulation of a linear system which provides the degree and the coefficients of the alcm using low rank approximation techniques and numerical optimization tools particularly in the presence of inaccurate data the numerical stability and complexity of the method are analysed and a comparison with other methods is provided

computing greatest common divisors and squarefree decompositions through matrix methods the parametric and approximate cases abstract barnett s method through bezoutians is a purely linear algebra method allowing to compute the degree of the greatest common divisor of several univariate polynomials in a very compact way two different uses of this method in computer algebra are introduced here firstly we describe an algorithm for parameterizing the greatest common divisor of several polynomials in k x y being x a parameter taking values in an real field k secondly we consider the problem of computing the approximate greatest common divisor with limited accuracy for several univariate polynomials following corless et al r m corless p m gianni b m trager s watt the singular value decomposition for polynomial systems in acm international symposium on symbolic and algebraic computation 1995 pp 195 207 given a family of polynomials whose coefficients are imperfectly known we describe an algorithm for computing their approximate greatest common divisor by using as main tools barnett s method and singular value decomposition computations furthermore we show how to use this algorithm in order to obtain the approximate squarefree decomposition of a given polynomial with imperfectly known coefficients

on the null spaces of the macaulay matrix abstract in this article both the left and right null space of the macaulay matrix are described the left null space is shown to be linked with the occurrence of syzygies in its row space it is also demonstrated how the dimension of the left null space is described by a piecewise function of polynomials we present two algorithms that determine these polynomials furthermore we show how the finiteness of the number of basis syzygies results in the notion of the degree of regularity this concept plays a crucial role in describing a basis for the right null space of the macaulay matrix in terms of differential functionals we define a canonical null space for the macaulay matrix in terms of the projective roots of a polynomial system and extend the multiplication property of this canonical basis to the projective case this results in an algorithm to determine the upper triangular commuting multiplication matrices finally we discuss how stetter s eigenvalue problem to determine the roots of a multivariate polynomial system can be extended to the case where a multivariate polynomial system has both affine roots and roots at infinity

structure of proofs and the complexity of cut elimination the importance of the structure of cut formulas with respect to proof length and proof depth has been studied in various occasions it has been illustrated that a quantifier may be more powerful than a binary connective in cut formulas with respect to the reduction or increase of proof length and proof depth and a sequence of quantifiers of the same type existential or universal may be less powerful than a sequence of quantifiers of alternating types this paper provides a refined view on cut elimination through an analysis of the structure of proofs brings new insight into the relation between cut formulas and short proofs and illustrates that a mixture of quantifiers and binary connectives could be important for achieving the maximal benefit of cut formulas for obtaining short proofs

periods of cusp forms and elliptic curves over imaginary quadratic fields in this paper we explore the arithmetic correspondence between on the one hand isogeny classes of elliptic curves e defined over an imaginary quadratic field k of class number one and on the other hand rational newforms f of weight two for the congruence subgroups where n is an ideal in the ring of integers r of k this continues work of the first author and forms part of the ph d thesis of the second author in each case we compute numerically the value of the l series at and compare with the value of which is predicted by the birch swinnerton dyer conjecture finding agreement to several decimal places in particular we find that whenever has a point of infinite order several examples are given in detail from the extensive tables computed by the authors

the calculation of the degree of an approximate greatest common divisor of two polynomials the calculation of the degree of an approximate greatest common divisor agcd of two inexact polynomials f y and g y is a non trivial computation because it reduces to the estimation of the rank loss of a resultant matrix r f g this computation is usually performed by placing a threshold on the small singular values of r f g but this method suffers from disadvantages because the numerical rank of r f g may not be defined or the noise level imposed on the coefficients of f y and g y may not be known or it may only be known approximately this paper addresses this problem by considering two methods for estimating the degree of an agcd of f y and g y such that knowledge of the noise level is not required the first method involves the calculation of the smallest angle between two subspaces that are apparent from the structure of the sylvester resultant matrix s f g and the second method uses the theory of subresultant matrices which are derived from s f g by the deletion of some of its rows and columns the two methods are compared computationally on non trivial polynomials

a geometrical approach to finding multivariate approximate lcms and gcds abstract in this article we present a new approach to compute an approximate least common multiple lcm and an approximate greatest common divisor gcd of two multivariate polynomials this approach uses the geometrical notion of principal angles whereas the main computational tools are the implicitly restarted arnoldi method and sparse qr decomposition upper and lower bounds are derived for the largest and smallest singular values of the highly structured macaulay matrix this leads to an upper bound on its condition number and an upper bound on the 2 norm of the product of two multivariate polynomials numerical examples are provided

asymptotic cyclic expansion and bridge groups of formal proofs abstract formal proofs even simple ones may hide an unexpected intricate combinatorics we define a new combinatorial invariant the bridge group of a proof which encodes the cyclic structure of proofs in the sequent calculus we compute the bridge groups of two infinite families of proofs and identify them with the baumslag solitar and gersten groups we observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs

binary hermitian forms over a cyclotomic field abstract let be a primitive fifth root of unity and let f be the cyclotomic field f q let o f be the ring of integers we compute the voronoi polyhedron of binary hermitian forms over f and classify gl 2 o conjugacy classes of perfect forms the combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group gl 2 o and hecke eigenforms