on the degrees of bases of free modulesover a polynomial ring let k be an infinite field a the polynomial ring k x 1 x n and f in a n times m a matrix such that rm im f subset a n is a free in particular quillen suslin theorem implies that rm ker f is also free let d be the maximum of the degrees of the entries of f and s the rank of f we show that there exists a basis v 1 ldots v m of a m such that v 1 ldots v m s is a basis of rm ker f f v m s 1 ldots f v m is a basis of rm im f and the degrees of their coordinates are of order m s sd o n 4

elimination ideals and bezout relations abstract let k be an infinite field and i k x 1 x n be a non zero ideal such that dim v i q 0 denote by f 1 f s a set of generators of i one can see that in the set i k x 1 x q 1 there exist non zero polynomials depending only on these q 1 variables we aim to bound the minimal degree of the polynomials of this type and of a bezout i e membership relation expressing such a polynomial as a combination of the f i in particular we show that if deg f i d i where d 1 d 2 d s then there exist a non zero polynomial x k x 1 x q 1 i such that deg d s i 1 n q 1 d i we also give a relative version of this theorem

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

multivariate mimo fir inverses the problem of computing exact finite impulse response fir inverses for multivariate multiple input multiple output mimo fir systems is considered necessary and sufficient conditions for invertibility are given along with computation techniques random systems and structured systems are defined necessary and sufficient conditions for the almost sure invertibility of structured random systems are derived bounds on the orders of the inverse filters are computed

on the effective nullstellensatz let mathbb k be an algebraically closed field and let x subset mathbb k m be an n dimensional affine variety assume that f1 fk are polynomials which have no common zeros on x we estimate the degrees of polynomials a i in mathbb k x such that 1 ki 1aifi on x our estimate is sharp for k n and nearly sharp for k n now assume that f1 fk are polynomials on x let i f 1 dots f k subset mathbb k x be the ideal generated by fi it is well known that there is a number e i the noether exponent such that ie i i we give a sharp estimate of e i in terms of n deg x and deg fi we also give similar estimates in the projective case finally we obtain a result from the elimination theory if f 1 dots f n in mathbb k x 1 dots x n is a system of polynomials with a finite number of common zeros then we have the following optimal elimination phi i x i sum n j 1 f jg ij quad i 1 dots n where deg f jg ij le prod n i 1 deg f i

a finite volume difference scheme for a model of settling particle dispersion from an elevated source in an open channel flow abstract a finite volume fitted difference scheme is constructed to solve the unsteady convective diffusion equation transformed on a finite domain modeling longitudinal dispersion of suspended particles with settling velocity in a turbulent shear flow over a rough bed surface first we discuss the well posedness of the differential problem and the non negativity of its solution then to overcome the degeneracy at the part of the boundary starting from the divergent form of the equation we perform a local fitted space discretization this approximation is determined by a set of two point boundary value problems non negativity of the numerical concentration of suspended fine particles is proved some results from computational experiments are presented to illustrate the properties of the constructed scheme

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

an a posteriori error estimate for finite element approximations of a singularly perturbed advection diffusion problem abstract in this paper the author presents an a posteriori error estimator for approximations of the solution to an advection diffusion equation with a non constant vector valued diffusion coefficient in a conforming finite element space based on the complementary variational principle we show that the error of an approximate solution in an associated energy norm is bounded by the sum of the weighted l 2 norms of solutions to a set of independent complementary variational problems each defined on only one element of the partition this error bound guarantees the over estimation of the true error and does not depend unfavourably on as goes to zero although the original equation is a non self adjoint problem the strong form of each local variational problem is always a poisson equation with neumann boundary conditions the approximation of these local problems is then discussed and it is shown that omitting a higher order term the finite element solutions of these local complementary variational problems provide a computable upper error bound for the original finite element approximation in the energy norm numerical results presented to validate the theoretical results show that the computed error bounds are tight for a wide range of values of and always over estimate the true errors

a second order time accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids a new second order time accurate fractional step method for solving unsteady incompressible navier stokes equations on hybrid unstructured grids is presented the nonstaggered grid method originally developed by rhie and chow 1983 aiaa j 21 1525 for steady flow and further extended by zang et al 1994 j comput phys 114 18 to unsteady flow on structured grids is employed in the present study to enforce mass conservation on hybrid unstructured grids the pressure and cartesian velocity components are defined at the center of each cell while the face normal velocities are defined at the mid points of the corresponding cell faces a second order fully implicit time advancement scheme is used for time integration and the resulting nonlinear equations are linearized without losing the overall time accuracy both the momentum and poisson equations are integrated with the finite volume method and the flow variables at the cell face are obtained using an interpolation scheme independent of cell shape the present numerical method is applied to four different benchmark problems and proves to be accurate and efficient

the finite volume method and application in combinations abstract in this paper the conventional finite volume method fvm is interpreted as a new kind of galerkin finite element method fem where the same piecewise linear functions are chosen as in both trial and test spaces and some specific integration rules are adopted error analysis is made for the regular delaunay triangulation involving obtuse triangles separated to prove optimal convergence rates of the approximate solutions obtained the new interpretation makes the fvm analysis much easier because we may bypass verification of the nontrivial ladyzhenskaya babuska brezzi lbb condition by the petrov galerkin fem in the existing analysis of fvm more importantly the new interpretation and the simple fvm analysis enable us to construct easily the combinations of fvm with other popular numerical methods such as the finite element method fem the finite difference method fdm the ritz galerkin method rgm etc for solving complicated problems of partial differential equations pde for example for solving singularity problems the combination of rgm fvm is superior to the combination of rgm fdm in flexibility of arbitrary solution domains and also superior to the combination of rgm fem in substantial saving of cpu time since the conservative law of flux may be maintained exactly in the numerical solutions and since obtuse triangles may be included in the delaunay triangulation the fvm and its combinations become very promising for solving elliptic boundary value problems in particular those where singularity solutions exist and those where the obeying of conservative law is crucial the numerical examples of combinations of rgm fvm are given for solving motz s problems to verify the optimal convergence rates the techniques and analysis of fvm and its combinations can be extended to the convection diffusion problems most importantly an important aspect in this paper is the possibility to include obtuse triangles in the error analysis

a subdivision based algorithm for the sparse resultant multivariate resultants generalize the sylvester resultant of two polynomials and characterize the solvability of a polynomial system they also reduce the computation of all common roots to a problem in linear algebra we propose a determinantal formula for the sparse resultant of an arbitrary system of n 1 polynomials in n variables this resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their bezout number our algorithm uses a mixed polyhedral subdivision of the minkowski sum of the newton polytopes in order to construct a newton matrix its determinant is a nonzero multiple of the sparse resultant and the latter equals the gcd of at most n 1 such determinants this construction implies a restricted version of an effective sparse nullstellensatz for an arbitrary specialization of the coefficients there are two methods that use one extra variable and yield the sparse resultant this is the first algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n we conjecture its extension to producing an exact rational expression for the sparse resultant

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

decomposition of residue currents given a submodule j o r 0 and a free resolution of j one can de ne a certain vector valued residue current whose annihilator is j we make a decomposition of the current with respect to assj that corresponds to a primary decomposition of j as a tool we introduce a class of currents that includes usual residue and principal value currents in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets

on bounds for the effective differential nullstellensatz abstract understanding bounds for the effective differential nullstellensatz is a central problem in differential algebraic geometry recently several bounds have been obtained using dicksonian and antichains sequences with a given growth rate in the present paper we make these bounds more explicit and therefore more applicable to understanding the computational complexity of the problem which is essential to designing more efficient algorithms

a bound for orders in differential nullstellensatz abstract we give the first known bound for orders of differentiations in differential nullstellensatz for both partial and ordinary algebraic differential equations this problem was previously addressed in a seidenberg an elimination theory for differential algebra univ of california publ in math iii 2 1956 31 66 but no complete solution was given our result is a complement to the corresponding result in algebraic geometry which gives a bound on degrees of polynomial coefficients in effective nullstellensatz g hermann die frage der endlich vielen schritte in der theorie der polynomideale math ann 95 1 1926 736 788 e w mayr a w meyer the complexity of the word problems for commutative semigroups and polynomial ideals adv math 46 3 1982 305 329 w d brownawell bounds for the degrees in the nullstellensatz ann of math 126 3 1987 577 591 j kollar sharp effective nullstellensatz j amer math soc 1 4 1988 963 975 l caniglia a galligo j heintz some new effectivity bounds in computational geometry in applied algebra algebraic algorithms and error correcting codes rome 1988 in lecture notes in comput sci vol 357 springer berlin 1989 pp 131 151 n fitchas a galligo nullstellensatz effectif et conjecture de serre theoreme de quillen suslin pour le calcul formel math nachr 149 1990 231 253 t krick l m pardo m sombra sharp estimates for the arithmetic nullstellensatz duke math j 109 3 2001 521 598 z jelonek on the effective nullstellensatz invent math 162 1 2005 1 17 t dube a combinatorial proof of the effective nullstellensatz j symbolic comput 15 3 1993 277 296 this paper is dedicated to the memory of eugeny pankratiev who was the advisor of the first three authors at moscow state university

residue calculus and effective nullstellensatz multivariate residue calculus in the spirit of j lipman is developed from the computational point of view for example with several variants of the classical transformation law and used in order to make totally explicit the bezout identity and therefore the algebraic nullstellensatz in k x 1 x n where k is an infinite field of arbitrary characteristic such identities provide sharp size estimates for the denominator and the divisors in the bezout identity when k is the quotient field of a factorial regular ring a equipped with a size such as z or f p 1 q the estimates obtained by the authors in a previous work in the case a z are sharpened here while analytic techniques are replaced with an algebraic approach

new effective differential nullstellensatz abstract we show new upper and lower bounds for the effective differential nullstellensatz for differential fields of characteristic zero with several commuting derivations seidenberg was the first to address this problem in 1956 without giving a complete solution in the case of one derivation the first bound is due to grigoriev in 1989 the first bounds in the general case appeared in 2009 in a paper by golubitsky kondratieva szanto and ovchinnikov with the upper bound expressed in terms of the ackermann function d alfonso jeronimo and solerno using novel ideas obtained in 2014 a new bound if restricted to the case of one derivation and constant coefficients to obtain the bound in the present paper without this restriction we extend this approach and use the new methods of freitag and leon sanchez and of pierce which represent a model theoretic approach to differential algebraic geometry

convergence of a fitted finite volume method for the penalized black scholes equation governing european and american option pricing in this paper we present an analysis of a numerical method for a degenerate partial differential equation called the black scholes equation governing american and european option pricing the method is based on a fitted finite volume spatial discretization and an implicit time stepping technique the analysis is performed within the framework of the vertical method of lines where the spatial discretization is formulated as a petrov galerkin finite element method with each basis function of the trial space being determined by a set of two point boundary value problems we establish the stability and an error bound for the solutions of the fully discretized system numerical results are presented to validate the theoretical results

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

a novel exponentially fitted triangular finite element method for an advection diffusion problem with boundary layers in this paper we develop an exponentially fitted finite element method for a singularly perturbed advection diffusion problem with a singular perturbation parameter this finite element method is based on a set of novel piecewise exponential basis functions constructed on unstructured triangular meshes the basis functions can not be expressed explicitly but the values of each of them and its associated flux at a point are determined by a set of two point boundary value problems which can be solved exactly a method for evaluating elements of the stiffness matrix is also proposed for the case that is small numerical results presented to validate the method show that the method is stable for a large range of it is also shown by the numerical results that the rate of convergence of the method in an energy norm is of orderh1 2when is small

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

fractional sylvester gallai theorems we prove fractional analogs of the classical sylvester gallai theorem our theorems translate local information about collinear triples in a set of points into global bounds on the dimension of the set specifically we show that if for every points v in a finite set there are at least v other points u v for which the line through v u contains a third point in v then the v resides in a 13 2 dimensional affine subspace of this result which is one of several variants we study is motivated by questions in theoretical computer science and in particular from the area of error correcting codes our proofs combine algebraic analytic and combinatorial arguments a key ingredient is a new lower bound for the rank of design matrices specified only by conditions on their zero non zero pattern

constraints on counterexamples to the casas alvero conjecture and a verification in degree 12 in the first theoretical part of this paper we prove a number of constraints on hypothetical counterexamples to the casas alvero conjecture building on ideas of graf von bothmer labs schicho and van de woestijne that were recently reinterpreted by draisma and de jong in terms of p adic valuations in the second computational part we present ideas improving upon diaz toca and gonzalez vega s grobner basis approach to the casas alvero conjecture one application is an extension of the proof of graf von bothmer et al to the cases 5p k 6p k and 7p k that is for each of these cases we determine the finite list of primes p to which their proof is not applicable finally by combining both parts we settle the casas alvero conjecture in degree 12 the smallest open case

a global smooth version of the classical łojasiewicz inequality abstract let f r n r be a function of class c d d 1 such that d f x 1 d 0 on r n then the following global ojasiewicz inequality holds true c d dist x f 0 f x 1 0 d f x for all x r n where c d d 2 2 d 1 and dist x a denotes the euclidean distance from x to a as applications of this inequality we have the following statements if the sets f 0 and f x 1 0 are non asymptotic at infinity then there exist positive constants e and r such that c d dist x f 0 d f x whenever dist x f 0 e and x r if f is a polynomial of degree d with an isolated critical point at the origin the following effective ojasiewicz inequality holds true c dist x f 0 d 2 d 3 n 1 2 f x for all x r for some c 0 and r 0 finally we establish a relation between the above global ojasiewicz inequality and the phenomenon of singularities at infinity

an analysis of a conforming exponentially fitted finite element method for a convection diffusion problem in this paper we present a convergence analysis for a conforming exponentially fitted galerkin finite element method with triangular elements for a linear singularly perturbed convection diffusion problem with a singular perturbation parameter e it is shown that the error for the finite element solution in the energy norm is bounded by o h e1 2 u 2 e 1 2 u 1 if a regular family of triangular meshes is used in the case that a problem contains only exponential boundary layers the method is shown to be convergent at a rate of h1 2 h ln e on anisotropic layer fitted meshes

a computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers singularly perturbed turning point problems tpps for second order ordinary differential equations des exhibiting twin boundary layers are considered in order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods in this method the given internal domain of definition of differential equation is divided into four subintervals and we solve the differential equation in each interval and combine the solution the implementation of the method on parallel architectures is mentioned error estimates are derived and numerical examples are given

the minimal components of the mayr meyer ideals rifi such that each ri has degree at most deg f kd n mayr and meyer in mm found generators of a family of ideals for which a doubly exponential bound in n is indeed achieved bayer and stillman bs showed that for these mayr meyer ideals any minimal generating set of syzygies has elements of doubly exponential degree in n koh k modified the original ideals to obtain homogeneous quadric ideals with doubly exponential syzygies and ideal membership equations bayer huneke and stillman asked whether the doubly exponential behavior is due to the number of minimal and or associated primes or to the nature of one of them this paper examines the minimal components and minimal primes of the mayr meyer ideals in particular in section 2 it is proved that the intersection of the minimal components of the mayr meyer ideals does not satisfy the doubly exponential property so that the doubly exponential behavior of the mayr meyer ideals must be due to the embedded primes the structure of the embedded primes of the mayr meyer ideals is examined in s2 there exist algorithms for computing primary decompositions of ideals see giannitrager zacharias gtz eisenbud huneke vasconcelos ehv or shimoyama yokoyama sy and they have been partially implemented on the symbolic computer algebra programs singular and macaulay2 however the mayr meyer ideals have variable degree and a variable number of variables over an arbitrary field and there are no algorithms to deal with this generality thus any primary decomposition of the mayr meyer ideals has to be accomplished with traditional proof methods small cases of the primary decomposition analysis were partially verified on macaulay2 and singular and the emphasis here is on partially the computers quickly run out of memory

sparse effective membership problems via residue currents we use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems our bounds depend on the volume of the newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems we present sparse versions of max nother s af bg theorem macaulay s theorem and kollar s effective nullstellensatz as well as recent results by hickel and andersson gotmark

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

multidimensional multichannel fir deconvolution using grobner bases we present a new method for general multidimensional multichannel deconvolution with finite impulse response fir convolution and deconvolution filters using grobner bases previous work formulates the problem of multichannel fir deconvolution as the construction of a left inverse of the convolution matrix which is solved by numerical linear algebra however this approach requires the prior information of the support of deconvolution filters using algebraic geometry and grobner bases we find necessary and sufficient conditions for the existence of exact deconvolution fir filters and propose simple algorithms to find these deconvolution filters the main contribution of our work is to extend the previous grobner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general fir filters the proposed algorithms obtain a set of fir deconvolution filters with a small number of nonzero coefficients a desirable feature in the impulsive noise environment and do not require the prior information of the support moreover we provide a complete characterization of all exact deconvolution fir filters from which good fir deconvolution filters under the additive white noise environment are found simulation results show that our approaches achieve good results under different noise settings

division formulas on projective varieties we introduce a division formula on a possibly singular projective subvariety x of complex projective space mathbb p n which e g provides explicit representations of solutions to various polynomial division problems on the affine part of x especially we consider a global effective version of the briancon skoda huneke theorem

the product of matrix subspaces abstract in factoring matrices into the product of two matrices operations are typically performed with elements restricted to matrix subspaces such modest structural assumptions are realistic for example in large scale computations this paper is concerned with analyzing associated matrix geometries curvature of the product of two matrix subspaces is assessed the case of vanishing curvature gives rise to the notion of factorizable matrix subspace this can be regarded as an analogue of the internal zappa szep product in group theory interpreted in this way several classical instances are encompassed by this structure the craig sakamoto theorem fits naturally into this framework

singularly perturbed boundary value problem for linear equations with turning points abstract in this paper the boundary value problems for a class of linear ordinary differential equations with turning points are studied under suitable assumptions the author proves the existence and uniqueness of solutions and obtains the uniformly valid asymptotic expansions of the solutions

global effective versions of the briancon skoda huneke theorem we prove global effective versions of the briancon skoda huneke theorem our results extend to singular varieties a result of hickel on the membership problem in polynomial ideals in cn and a related theorem of ein and lazarsfeld for smooth projective varieties the proofs rely on known geometric estimates and new results on multivariable residue calculus

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

a global lojasiewicz inequality for algebraic varieties let x be the locus of common zeros of polynomials f 1 f in n complex variables a global upper bound for the distance to x is given in the form of a lojasiewicz inequality the exponent in this inequality is bounded by d min n k where d max 3 deg f i the estimates are also valid over an algebraically closed field of any characteristic

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

solving convection dominated anisotropic diffusion equations by an exponentially fitted finite volume method abstract this paper deals with the numerical solution by an exponentially fitted finite volume method of a linear convection dominated diffusion equation with a constant matrix diffusion coefficient and a singular perturbation parameter e the equation is first transformed into one with a scalar diffusion coefficient by a simple coordinate transformation and an exponentially fitted finite volume method is then applied to the resulting equation application of this approach to the time space discretisation of time dependent singularly perturbed convection diffusion equations is also discussed numerical examples are solved to demonstrate the usefulness of this approach the numerical results show that the method is efficient and stable for a large range of values of e

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

on counting integral points in a convex rational polytope given a convex rational polytope b x r n ax b we consider the functionb f b which counts the nonnegative integral points of b a closed form expression of its z transformz f z is easily obtained so thatf b can be computed as the inverse z transform of f we then provide two variants of an inversion algorithm as a by product one of the algorithms provides the ehrhart polynomial of a convex integer polytope we also provide an alternative that avoids the complex integration of f z and whose main computational effort is to solve a linear system this latter approach is particularly attractive for relatively small values ofm wherem is the number of nontrivial constraints or rows ofa

a nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection diffusion equation in this paper we formulate and analyze a discretization method for a 2d linear singularly perturbed convection diffusion problem with a singular perturbation parameter e the method is based on a nonconforming combination of the conventional galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method and on a mixture of triangular and rectangular elements it is shown that the method is stable with respect to a semi discrete energy norm and the approximation error in the semi discrete energy norm is bounded by ch ln e ln h with c independent of the mesh parameter h the diffusion coefficient e and the exact solution of the problem

on the number of zero patterns of a sequence of polynomials let f f1 fm be a sequence of polynomials of degree d in n variables m n over a field f the zero pattern of f at u f n is the set of those i 1 i m for which fi u 0 let zf f denote the number of zero patterns of f as u ranges over f we prove that zf f pn j 0 m j for d 1 and 1 zf f md n for d 2 for m nd these bounds are optimal within a factor of 7 25 the bound 1 improves the bound 1 md proved by j heintz 1983 using the dimension theory of affine varieties over the field of real numbers bounds stronger than heintz s but slightly weaker than 1 follow from results of j milnor 1964 h e warren 1968 and others their proofs use techniques from real algebraic geometry in contrast our half page proof is a simple application of the elementary linear algebra bound heintz applied his bound to estimate the complexity of his quanti fier elimination algorithm for algebraically closed fields we give sev eral additional applications the first two establish the existence of certain combinatorial objects our first application motivated by the branching program model in the theory of computing asserts that over any field f most graphs with v vertices have projective dimension p v log v the implied constant is absolute this result was previ ously known over the reals pudlak rodl the second application con cerns a lower bound in the span program model for computing boolean functions the third application motivated by a paper by n alon gives nearly tight ramsey bounds for matrices whose entries are defined by zero patterns of a sequence of polynomials we conclude the paper with a number of open problems computer and automation research institute hungarian academy of sci ences h 1111 budapest lagymanyosi u 11 hungary e mail address lajos nyest ilab sztaki hu department of computer science university of chicago chicago il 60637 e mail address laci cs uchicago edu department of computer science university of chicago chicago il 60637 e mail address gmkrishn cs uchicago edu

representations of integers by systems of three quadratic forms it is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers n1 nr by a system of quadratic forms q1 qr in k variables as long as k is sufficiently large with respect to r reducing the required number of variables remains a significant open problem in this work we consider the case of three forms and improve on the classical result by reducing the number of required variables to k 10 for almost all tuples under a non singularity assumption on the forms q1 q2 q3 to accomplish this we develop a three dimensional analogue of kloosterman s circle method in particular capitalizing on geometric properties of appropriate systems of three quadratic forms

the finite element method with weighted basis functions for singularly perturbed convection diffusion problems in this paper we present a finite element method for singularly perturbed convection diffusion problems in both one and two dimensions based on a set of weighted basis functions constructed on unstructured meshes in 2d for the one dimensional case both first and second order schemes are discussed a technique for approximating fluxes is proposed some theoretical results on uniform convergence are obtained for the two dimensional case a first order scheme is constructed for problems with two singular perturbation parameters a technique is also developed in approximating fluxes in 2d this technique is used to simplify the calculation of the integrals in the stiffness matrix arising from the scheme which will save computational costs the numerical results support the theoretical results and demonstrate that the method is stable for a wide range of singular perturbation parameters

an exponentially fitted finite volume method for the numerical solution of 2d unsteady incompressible flow problems abstract in this paper we develop and test an exponentially fitted finite volume method for the numerical solution of the navier stokes equations describing 2d incompressible flows the method is based on an unstructured delaunay mesh and its dual dirichlet tessellation combined with a locally constant approximation to the flux this yields a piecewise exponential approximation to the exact solution numerical tests are presented for a linear advection diffusion problem with boundary layers the method is then applied to the driven cavity problem with reynolds numbers up to 10 4 the numerical results indicate that the method is robust for a wide range of values of the reynolds number in the case re 10 4 unsteady solutions are captured if the mesh is sufficiently fine

explicit representation of membership in polynomial ideals we introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates we consider simple examples as the classical macaulay theorem as well as a quite recent result by hickel related to the effective nullstellensatz we also obtain a related result that generalizes max noether s classical af bg theorem

proof mining and effective bounds in differential polynomial rings abstract using the functional interpretation from proof theory we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings we extract effective bounds some of which are new to the literature from the resulting proofs in the process we discuss the constructive content of noetherian rings and the nullstellensatz in both the classical and differential settings sufficient background is given to understand the proof theoretic and differential algebraic framework of the main results

on degree bounds for the sparse nullstellensatz abstract we prove new upper bounds for the degrees in hilbert s nullstellensatz and for the noether exponent of polynomial ideals in terms of the monomial structure of the polynomials involved our bounds improve the previously known bounds in the sparse setting and are the first to take into account the different supports of the polynomials

łojasiewicz inequality at infinity for polynomials in two real variables we propose different types of ojasiewicz inequality at infinity for polynomials in two real variables the formulas for the ojasiewicz exponents are given

index of nilpotency of binomial ideals abstract in this paper we study and compute bounds for the index of nilpotency of lattice and cellular binomial ideals which are optimal in many cases this computations can be generalized to binomial ideals getting an effective hilbert s nullstellensatz for binomial ideals

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem