asymptotic expansion of complex integrals via mellin transform soit x une variete complexe connexe a n 1 dimensions et f une fonction holomorphe non constante sur x avec 0 comme seule valeur critique possible on demontre un theoreme de developpement asymptotique d integrales complexes par la transformation de mellin

irreducible affine space curves and the uniform artin rees property on the prime spectrum abstract we give a negative answer to the problem open for twenty years as to whether the full uniform artin rees property holds on the prime spectrum of an excellent ring it was known to hold locally on the prime spectrum of such a ring

lech s inequality the stuckrad vogel conjecture and uniform behavior of koszul homology abstract let r m be a noetherian local ring and let m be a finitely generated r module of dimension d we prove that the set l m i m e i m i m is bounded below by 1 d e r where r r ann m moreover when m is equidimensional this set is bounded above by a finite constant depending only on m the lower bound extends a classical inequality of lech and the upper bound answers a question of stuckrad vogel in the affirmative as an application we obtain results on uniform behavior of the lengths of koszul homology modules

regularizations of residue currents under assumptions about complete intersection we prove that coleff herrera type currents satisfy a robust calculus in the sense that natural regularizations of such currents can be multiplied to yield regularizations of the coleff herrera product of the currents

regularizations of products of residue and principal value currents let f 1 and f 2 be two functions on some complex n manifold and let be a test form of bidegree n n 2 assume that f 1 f 2 defines a complete intersection the integral of f 1 f 2 on f 1 2 e lunate 1 f 2 2 e lunate 2 is the residue integral i f1 f 2 e lunate 1 e lunate 2 it is in general discontinuous at the origin let 1 and 2 be smooth functions on 0 such that j 0 0 and j 1 we prove that the regularized residue integral defined as the integral of over 1 over 2 f 1 f 2 where j j f j 2 e lunate j is holder continuous on the closed first quarter and that the value at zero is the coleff herrera residue current acting on in fact we prove that if is a test form of bidegree n n 1 then the integral of 1 over 2 f 1 f 2 is holder continuous and tends to the over potential 1 f 1 over 1 f 2 of the coleff herrera current acting on more generally let f 1 and f 2 be sections of some vector bundles and assume that f 1 f 2 defines a complete intersection there are associated principal value currents u f and u g and residue currents r f and r g the residue currents equal the coleff herrera residue currents locally one can give meaning to formal expressions such as e g u f r g in such a way that formal leibnitz rules hold our results generalize to products of these currents as well 2006 elsevier inc all rights reserved

explicit serre duality on complex spaces abstract in this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization as well as a new analytic proof of serre duality on any reduced pure n dimensional paracompact complex space x at the core of the paper is the introduction of certain fine sheaves b x n q of currents on x of bidegree n q such that the dolbeault complex b x n becomes in a certain sense a dualizing complex in particular if x is cohen macaulay then b x n is an explicit fine resolution of the grothendieck dualizing sheaf

integral transforms of analytic functions on abstract wiener spaces abstract let h b be an abstract wiener pair and pt the wiener measure with variance t let e a be the class of exponential type analytic functions defined on the complexification b of b for each pair of nonzero complex numbers and f e a we define f f y b f x y p 1 dx y b we show that the inverse f 1 exists and there exist two nonzero complex numbers such that f 1 f 1 1 clearly the fourier wiener transform the fourier feynman transform and the gauss transform are special cases of f finally we apply the transform to investigate the existence of solutions for the differential equations associated with the operator n c where c is a nonzero complex number and n c is defined by n c u x u x c x du x where is the laplacian and is the b b pairing we show that the solutions can be represented as integrals with respect to the wiener measure

new upper bounds on error exponents we derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels this is an improvement on the best earlier known bounds by shannon gallager berlekamp 1967 and mceliece omura 1977 for the probability of undetected error the new bounds are better than the bounds by levenshtein 1978 1989 and the bound by abdel ghaffar see ibid vol 43 p 1489 502 1997 moreover we further extend the range of rates where the undetected error exponent is known to be exact the new bounds are based on an analysis of possible distance distributions of the codes along with some inequalities relating the distance distributions to the error probabilities

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

abelian differentials on singular varieties and variations on a theorem of lie griffiths we compare various notions of meromorphic and holomorphic differential forms on singular analytic varieties in particular we find that every meromorphic form gives rise to a canonical principal value current with support in the variety demanding that this current be closed we obtain a useful analytic description of the dualizing sheaf we then go on to generalize the lie griffiths theorem on one hand it is shown that a rational trace is enough to ensure algebraicity of the data on the other hand we prove that zero trace implies that the form is holomorphic in the sense of currents in fact we prove a more general local version of the theorem

the bar partial equation on a non reduced analytic space let x be a possibly non reduced analytic space of pure dimension we introduce a notion of overline partial equation on x and prove a dolbeault grothendieck lemma we obtain fine sheaves mathcal a x q of 0 q currents so that the associated dolbeault complex yields a resolution of the structure sheaf mathscr o x our construction is based on intrinsic semi global koppelman formulas on x

various approaches to products of residue currents abstract we describe various approaches to coleff herrera products of residue currents r j of cauchy fantappie leray type associated to holomorphic mappings f j more precisely we study to which extent exterior products of natural regularizations of the individual currents r j yield regularizations of the corresponding coleff herrera products our results hold globally on an arbitrary pure dimensional complex space

residue currents with prescribed annihilator ideals on singular varieties given an ideal mathcal j on a complex manifold andersson and wulcan constructed explicitly a current r mathcal j such that the annihilator of r mathcal j is mathcal j generalizing the duality theorem for coleff herrera products we describe a way to generalize this construction to ideals on singular varieties

generalized functions on infinite dimensional spaces and its application to white noise calculus in this paper a theory of generalized functions is established on an arbitrary abstract wiener space it is shown that it provides a nice framework for reformulating white noise calculus initiated by t hida unlike hida s approach the generalized brownian functionals are defined and studied through the generalized function test function pairing instead of employing the u functionals besides the fourier transform is extended to infinite dimensions and a new definition of the composition of a tempered distribution with the brownian motion is given at the end of the paper the ito formula for generalized brownian functionals and the kubo takenaka s theorem for stochastic integration are verified by our approach

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

lemme d artin rees theoreme d izumi et fonction de artin abstract we interpret the artin rees lemma and the izumi theorem in term of artin function and we obtain a stable version of the artin rees lemma we present different applications of these interpretations first we show that the artin function of x 1 x 2 x 3 x 4 as a polynomial in the ring of power series in more than three variables is not bounded by an affine function then we prove that the artin functions of a class of polynomials are bounded by affine functions and we use this to compute approximated integral closures of ideals

uniform annihilators of local cohomology of excellent rings abstract the main purpose of this paper is to give an affirmative answer to a conjecture of huneke c huneke uniform bounds in noetherian rings invent math 107 1992 203 223 conjecture 2 13

a briancon skoda theorem for isolated singularities abstract letabe a normal cohen macaulay local ring which is a homomorphic image of a gorenstein local ring supposing thatais pseudo rational in the punctured spectrum and that there exists a proper birational morphismx spec asuch thatxis pseudo rational we show how one can find an integerk 0 such that i n k infor all idealsi aand integersn 0

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

a generalization of the riesz representation theorem to infinite dimensions lethbe a real separable hilbert space and lete hbe a nuclear space with the chain of hilbert spaces ep p 1 2 3 such thate p 1 ep lete ande pdenote the dual spaces ofeandep respectively let cpbe the collection of real valued functionsfdefined one psuch thatfis uniformly continuous on bounded subsets ofe pand such that f p supx e p f x exp 12 x 2 p is finite set c p 1 cp thenc is a complete countably normed space equipped with the family p p 1 2 3 of norms in this paper it is shown that to every bounded linear functionalfin c there corresponds a signed measure fsuch thatf e x f dx for c it is also shown that there exists somepsuch that the measurable support of is contained ine pand e p exp 12 x 2 p f dx this result extends the riesz representation theorem to infinite dimensions in the course of the proof an infinite dimensional analogue of the weierstrass approximation theorem is also established one

on steinʼs method for infinite dimensional gaussian approximation in abstract wiener spaces abstract in this paper we generalize stein s method to infinite variate normal approximation that is an infinite dimensional approximation by abstract wiener measures on a real separable banach space we first establish a stein s identity for abstract wiener measures and solve the corresponding stein s equation then we will present a gaussian approximation theorem using exchangeable pairs in an infinite variate context as an application we will derive an explicit error bound of gaussian approximation to the distribution of a sum of independent and identically distributed banach space valued random variables based on a lindeberg levy type limit theorem in addition an analogous of berry esseen type estimate for abstract wiener measures will be obtained

analytic version of test functionals fourier transform and a characterization of measures in white noise calculus it is shown that the space j of test white noise functionals has an analytic version a which is an algebra as well as a topological linear space topologized by the projective limit of a sequence ap p n of banach spaces with respective norm given by ap supz cj p z exp 2 1 z p2 where sj p denotes the complexification of j p furthermore it is shown that the a topology and the j topology are equivalent in the course of the proof it is also shown that the space jp is isometrically isomorphic to the bargmann segal analytic functions on bj p under s transform employing this new version we are able to define the fourier transform of a generalized white noise functional as the adjoint of fourier wiener transform t1 i with parameter 1 i and moreover we show that every measure in j always satisfies a certain growth condition

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

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

integral degree of a ring reduction numbers and uniform artin rees numbers abstract the supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree a new invariant of the ring which is finite provided the ring has finite integral closure as a consequence one obtains bounds for the castelnuovo mumford regularity of the rees algebra and for the artin rees numbers

on the duality theorem on an analytic variety the duality theorem for coleff herrera products on a complex manifold says that if f f 1 f p defines a complete intersection then the annihilator of the coleff herrera product f equals locally the ideal generated by f this does not hold unrestrictedly on an analytic variety z we give necessary and in many cases sufficient conditions for when the duality theorem holds these conditions are related to how the zero set of f intersects certain singularity subvarieties of the sheaf mathcal o z

the bee identification problem bounds on the error exponent consider the problem of identifying a massive number of bees uniquely labeled with barcodes using noisy measurements we formally introduce this bee identification problem define its error exponent and derive efficiently computable upper and lower bounds for this exponent we show that joint decoding of barcodes provides a significantly better exponent compared to separate decoding followed by permutation inference for low rates we prove that the lower bound on the bee identification exponent obtained using typical random codes trc is strictly better than the corresponding bound obtained using a random code ensemble rce further as the rate approaches zero we prove that the upper bound on the bee identification exponent meets the lower bound obtained using trc with joint barcode decoding

the bee identification error exponent with absentee bees the bee identification problem was formally defined by tandon tan and varshney ieee trans commun vol 67 2019 and the error exponent was studied this work extends the results for the absentee bees scenario where a fraction of the bees are absent in the beehive image used for identification for this setting we present an exact characterization of the bee identification error exponent and show that independent barcode decoding is optimal i e joint decoding of the bee barcodes does not result in a better error exponent relative to independent decoding of each noisy barcode this is in contrast to the result without absentee bees where joint barcode decoding results in a significantly higher error exponent than independent barcode decoding we also define and characterize the capacity for the bee identification problem with absentee bees and prove a strong converse for the same

sharp inequalities and regularity of heat semigroup on infinite dimensional spaces abstract this paper discusses the regularity of the heat semigroup p t generated from the abstract wiener measure p t on an abstract wiener space h b it is shown that if f l x p t x dy for some 1 then p t f x is infinitely h differentiable and d n p t f x is a symmetric n linear hilbert schmidt operator in l n h a compact formula of d n p t f x and the estimation of d n p t f x hs are also given in the course of proving the above results we obtain some sharp integral inequalities

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

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

uniform annihilators of local cohomology of extended rees rings abstract the purpose of the paper is to study the annihilators of local cohomology of the extended rees rings we will prove that if a m is a noetherian local ring with infinite residue field and a is a uniform local cohomological annihilator of a then there is an integer n such that a t n is a uniform local cohomological annihilator of the extended rees ring a t t 1 i for any m primary ideal i of a

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by y x in this paper the analogous result is shown for x and y defined over a perfect field of finite characteristic moreover a local construction of y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

total separable closure and contractions abstract we show that on integral normal separated schemes whose function field is separably closed for each pair of points the intersection of the resulting local schemes is local this extends a result of artin from rings to schemes the argument relies on the existence of certain modifications in inverse limits as an application we show that cech cohomology coincides with sheaf cohomology for the nisnevich topology along the way we generalize the characterization of contractible curves on surfaces by negative definiteness of the intersection matrix to higher dimensions using bigness of invertible sheaves on non reduced schemes

uniform artin rees bounds for syzygies abstract let r m be a local noetherian ring let m be a finitely generated r module and let f be a free resolution of m we find a uniform bound h such that the artin rees containment i n f i im i 1 i n h im i 1 holds for all integers i d for all integers n h and for all ideals i of r in fact we show that a considerably stronger statement holds the uniform bound h holds for all ideals and all resolutions of dth syzygy modules in order to prove our statements we introduce the concept of koszul annihilating sequences

castelnuovo mumford regularity postulation numbers and relation types abstract we establish a bound for the castelnuovo mumford regularity of the associated graded ring g i a of an m primary ideal i of a local noetherian ring a m in terms of the dimension of a the relation type and the number of generators of i as a consequence we obtain that the existence of uniform bounds for the regularity of the associated graded ring and the relation type of parameter ideals in a are equivalent conditions in addition we establish an equation for the postulation number and the castelnuovo mumford regularity of the associated graded ring g q a of a parameter ideal q which holds under certain conditions on the depths of the occurring rings we also show that the regularity of the ring g q a is bounded in terms of the dimension of a the length of a q and the postulation number of g q a

uniform annihilators of local cohomology abstract in this paper we study the properties of noetherian rings containing uniform local cohomological annihilators it turns out that all such rings should be universally catenary and locally equidimensional we will prove a necessary and sufficient condition for such rings which enables us to show that if a locally equidimensional ring r is the image of a cohen macaulay ring then r has a uniform local cohomological annihilator moreover we will give a positive answer to a conjecture of huneke c huneke uniform bounds in noetherian rings invent math 107 1992 203 223 conjecture 2 13 about excellent rings with dimension no more than 5

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

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

a dolbeault grothendieck lemma on complex spaces via koppelman formulas let x be a complex space of pure dimension we introduce fine sheaves a x q of 0 q currents which coincides with the sheaves of smooth forms on the regular part of x so that the associated dolbeault complex yields a resolution of the structure sheaf hol x our construction is based on intrinsic and quite explicit semi global koppelman formulas

error exponents of typical random codes we define the error exponent of the typical random code trc as the long block limit of the negative normalized expectation of the logarithm of the error probability of the random code as opposed to the traditional random coding error exponent which is the limit of the negative normalized logarithm of the expectation of the error probability for the ensemble of uniformly randomly drawn fixed composition codes we provide exact error exponents of trcs for a general discrete memoryless channel and a wide class of stochastic decoders collectively referred to as the generalized likelihood decoder gld this ensemble of fixed composition codes is shown to be no worse than any other ensemble of independent codewords that are drawn under a permutation invariant distribution e g independent identically distributed codewords we also present relationships between the error exponent of the trc and the ordinary random coding error exponent as well as the expurgated exponent for the gld finally we demonstrate that our analysis technique is applicable also to more general communication scenarios such as list decoding for fixed size lists as well as the decoding with an erasure list option in forney s sense

the kashiwara conjugation and wave front sets of regular holonomic distributions on a complex manifold if a distributionu on a complex manifoldx is a solution of a regular holonomicdx module one shows that its wave front sets coincide with the characteristic variety of thedx module generated byu the proof relies essentially on a microlocal form of kashiwara s conjugation functor as a byproduct regular holonomicex modules are locally embedded into the sheaf of temperate microfunctions