splittable ideals and the resolutions of monomial ideals abstract we provide a new combinatorial approach to study the minimal free resolutions of edge ideals that is quadratic square free monomial ideals with this method we can recover most of the known results on resolutions of edge ideals with fuller generality and at the same time obtain new results past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or koszul homology groups our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph we also show that our technique extends successfully to the study of graded betti numbers of arbitrary square free monomial ideals viewed as facet ideals of simplicial complexes

ideals with stable betti numbers componentwise linear ideals were introduced earlier to generalize the result that the stanley reisner ideal i of a simplicial complex has a linear resolution if and only if its alexander dual is cohen macaulay it turns out that i is componentwise linear if and only if is sequentially cohen macaulay in this paper we discuss betti number properties of componentwise linear ideals let i be a graded ideal of a polynomial ring s and gin i the generic initial ideal of i with respect to the reverse lexicographic term order on s our main result is that i and gin i admit the same graded betti numbers if and only if i is componentwise linear for the proof of this fact we describe some properties of the betti diagram of a generic initial ideal combinatorial implications for shifted complexes will also be discussed

maximal minimal resolutions that is exact everywhere except at f0 where the cokernel is m such a resolution always exists if r is a noetherian n graded ring with r0 a field and m is a finitely generated z graded r module then a resolution may be constructed in a minimal way in which each fi is graded and each differential is homogeneous of degree 0 the rank of the ith free module in a minimal free resolution of m is an invariant of m called the ith betti number of m and denoted i m likewise r ij m the number of elements of degree j in a minimal set of homogeneous generators of fi is also an invariant of m called the i j th graded betti number of m in this paper we study modules with maximal graded betti numbers more precisely consider a set consisting of pairs r m where r is a noetherian n graded ring with r0 a field and m is a finitely generated graded r module is there some r m such that ij m ij m for every i and j and every r m there are trivial examples in which the answer is yes e g has only one element or no e g consists of all such pairs theorems 1 and 2 below give affirmative answers for certain sets defined by conditions on the hilbert series and the depths of r and m theorem 3 allows one to compute the maximal graded betti numbers for the pairs in the sets considered in theorems 1 and 2 throughout this paper k denotes a field and q the polynomial ring k x1 xn with the usual n grading given by degxi 1 the hilbert series of a finitely generated graded q module m is hm s dimk mis we define d lexicographic ideals and submodules in section 2 the poincare series of an r module m is p m s t ij m s t we write 4 for coefficient wise inequality of laurent series with coefficients in z

symbolic powers of sums of ideals let i and j be nonzero ideals in two noetherian algebras a and b over a field k let i j denote the ideal generated by i and j in a otimes k b we prove the following expansion for the symbolic powers begin aligned i j n sum i j n i i j j end aligned if a and b are polynomial rings and if mathrm char k 0 or if i and j are monomial ideals we give exact formulas for the depth and the castelnuovo mumford regularity of i j n which depend on the interplay between the symbolic powers of i and j the proof involves a result of independent interest which states that the induced map mathrm tor i a k i n rightarrow mathrm tor i r k i n 1 is zero for any homogeneous ideal i and i ge 0 n ge 0 we also investigate other properties and invariants of i j n such as the equality between ordinary and symbolic powers the waldschmidt constant and the cohen macaulayness

betti splitting via componentwise linear ideals abstract a monomial ideal i admits a betti splitting i j k if the betti numbers of i can be determined in terms of the betti numbers of the ideals j k and j k given a monomial ideal i we prove that i j k is a betti splitting of i provided j and k are componentwise linear generalizing a result of francisco ha and van tuyl if i has a linear resolution the converse also holds we apply this result recursively to the alexander dual of vertex decomposable shellable and constructible simplicial complexes moreover we determine the graded betti numbers of the defining ideal of three general fat points in the projective space

smallest graded betti numbers it is known that given a hilbert function h there need not exist a module which has uniquely the smallest graded betti numbers among all modules attaining h in this paper we extend the previous example of this behavior to an infinite family and demonstrate with a second infinite family that even when the given hilbert function is that of a complete intersection a module with uniquely smallest graded betti numbers need not exist finally we prove a conjecture of geramita harima and shin concerning the non existence of uniquely smallest graded betti numbers among all gorenstein rings attaining a given hilbert function

fixing monotone boolean networks asynchronously the asynchronous automaton associated with a boolean network f f0 1gn f0 1gn is considered in many applications it is the nite deterministic automaton with set of states f0 1gn alphabet f1 ng where the action of letter i on a state x consists in either switching the ith component if fi x 6 xi or doing nothing otherwise this action is extended to words in the natural way we then say that a word w xes f if for all states x the result of the action of w on x is a xed point of f in this paper we ask for the existence of xing words and their minimal length firstly our main results concern the minimal length of words that x monotone networks we prove that for n suciently large there exists a monotone network f with n components such that any word xing f has length n2 for this rst result we prove using baranyai s theorem a property about shortest supersequences that could be of independent interest there exists a set of permutations of f1 ng of size 2o n such that any sequence containing all these permutations as subsequences is of length n2 conversely we construct a word of length o n3 that xes all monotone networks with n components secondly we rene and extend our results to dierent classes of xable networks including networks with an acyclic interaction graph increasing networks conjunctive networks monotone networks whose interaction graphs are contained in a given graph and balanced networks

nonreversibility and equivalent constructions of multiple unicast networks we prove that for any finite directed acyclic network there exists a corresponding multiple unicast network such that for every alphabet each network is solvable if and only if the other is solvable and for every finite field alphabet each network is linearly solvable if and only if the other is linearly solvable the proof is constructive and creates an extension of the original network by adding exactly s 5m r 1 new nodes where in the original network m is the number of messages r is the average number of receiver nodes demanding each source message and s is the number of messages emitted by more than one source the construction is then used to create a solvable multiple unicast network which becomes unsolvable over every alphabet size if all of its edge directions are reversed and if the roles of source receiver pairs are reversed

artinian level modules and cancellable sequences in order to use dualization to study hilbert functions of artinian level algebras we extend the notion of level sequences and cancellable sequences introduced by geramita and lorenzini to include

on some gorenstein loci in hilb_6 p 4_k abstract let k be an algebraically closed field and let h ilb d a g p k d 2 be the open locus inside the hilbert scheme h ilb d p k d 2 corresponding to arithmetically gorenstein subschemes we prove the irreducibility and characterize the singularities of h ilb 6 a g p k 4 in order to achieve these results we also classify all artinian gorenstein not necessarily graded k algebras up to degree 6 moreover we describe the loci in h ilb 6 a g p k 4 obtained via some geometric construction finally we prove the obstructedness of some families of points in h ilb d a g p k d 2 for each d 6

k configurations in p3all have extremal resolutions abstract we prove that allk configurations of points in p 3have the extremal resolution for a cohen macaulay scheme with the hilbert function

ideals containing the squares of the variables we study the betti numbers of graded ideals containing the squares of the variables in a polynomial ring we prove the lex plus powers conjecture for such ideals

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

further bibasic hypergeometric transformations and their applications abstract in this paper we give several new transformation formulae and generalize one result obtained by singh u b singh certain bibasic hypergeometric transformations and their applications j math anal appl 201 1996 44 56 with the help of bailey s transform further some new multiple series identities of the rogers ramanujan type are established

on borel fixed ideals generated in one degree abstract we construct a shellable polytopal cell complex that supports a minimal free resolution of a borel fixed ideal which is borel generated by just one monomial in s k x 1 x 2 x n this includes the case of powers of the homogeneous maximal ideal x 1 x 2 x n as a special case in our most general result we prove that for any borel fixed ideal i generated in one degree there exists a polytopal cell complex that supports a minimal free resolution of i

pruned cellular free resolutions of monomial ideals abstract using discrete morse theory we give an algorithm that prunes the excess of information in the taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal the pruned resolution is not simplicial in general but we can slightly modify our algorithm in order to obtain a simplicial resolution we also show that the lyubeznik resolution fits into our pruning strategy the pruned resolution is not always minimal but it is a lot closer to the minimal resolution than the taylor and the lyubeznik resolutions as we will see in some examples we finally use our methods to give a different approach to the theory of splitting of monomial ideals we deduce from this splitting strategy that the pruned resolution is always minimal in the case of edge ideals of paths and cycles

nonunimodal gorenstein sequences of higher socle degrees abstract we study gorenstein h vectors h 0 h 1 h e of socle degree e with h 1 h i for each i and find a necessary and sufficient condition that there exists a nonunimodal gorenstein sequence in terms of socle degree e and codimension h 1 in particular we prove that there exist nonunimodal gorenstein h vectors if and only if h 1 4 e 3 for e 4 we also find infinitely many cases of non gorenstein h vectors having the lower bound in 15 this result generalizes the recent work 17 that the h vector 1 12 11 12 1 is not a gorenstein sequence

algebra structures on some canonical resolutions the study of algebra structures on finite free resolutions of cyclic modules begins with buchsbaum and eisenbud b el the classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field gu complete intersections koszul complex modules of homological dimension atmost 3 b el gorenstein of codimension four k m and herzog algebras k m23 avramov al gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure let r be a noetherian local ring with maximal ideal m in this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules r i namely when z is of the form jk where j is an ideal generated by a regular sequence and when z is the ideal of maximal minors of a generic n x m matrix provided r contains the rationals in al avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non zero obstructions the general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module when z is an ideal generated by a regular sequence in r and m r z for any positive integer k then these obstructions are all zero therefore avramov and schlessinger asked whether the minimal resolutions of r z admit an algebra structure corollary 3 6 of this paper provides an affirmative answer to this question

on the first infinitesimal neighborhood of a linear configuration of points in p2 abstract we consider the following open questions fix a hilbert function h that occurs for a reduced zero dimensional subscheme of p 2 among all subschemes x with hilbert function h what are the possible hilbert functions and graded betti numbers for the first infinitesimal neighborhood z of x i e the double point scheme supported on x is there a minimum h min and maximum h max such function the numerical information encoded in h translates to a type vector which allows us to find unions of points on lines called linear configurations with hilbert function h we give necessary and sufficient conditions for the hilbert function and graded betti numbers of the first infinitesimal neighborhoods of all such linear configurations to be the same even for those h for which the hilbert functions or graded betti numbers of the resulting double point schemes are not uniquely determined we give one depending only on h that does occur we prove the existence of h max in general and discuss h min our methods include liaison techniques

networks matroids and non shannon information inequalities we define a class of networks called matroidal networks which includes as special cases all scalar linearly solvable networks and in particular solvable multicast networks we then present a method for constructing matroidal networks from known matroids we specifically construct networks that play an important role in proving results in the literature such as the insufficiency of linear network coding and the unachievability of network coding capacity we also construct a new network from the vamos matroid which we call the vamos network and use it to prove that shannon type information inequalities are in general not sufficient for computing network coding capacities to accomplish this we obtain a capacity upper bound for the vamos network using a non shannon type information inequality discovered in 1998 by zhang and yeung and then show that it is smaller than any such bound derived from shannon type information inequalities this is the first application of a non shannon type inequality to network coding we also compute the exact routing capacity and linear coding capacity of the vamos network finally using a variation of the vamos network we prove that shannon type information inequalities are insufficient even for computing network coding capacities of multiple unicast networks

reduced arithmetically gorenstein schemes and simplicial polytopes with maximal betti numbers abstract an si sequence is a finite sequence of positive integers which is symmetric unimodal and satisfies a certain growth condition these are known to correspond precisely to the possible hilbert functions of graded artinian gorenstein algebras with the weak lefschetz property a property shared by a nonempty open set of the family of all graded artinian gorenstein algebras having a fixed hilbert function that is an si sequence starting with an arbitrary si sequence we construct a reduced arithmetically gorenstein configuration g of linear varieties of arbitrary dimension whose artinian reduction has the given si sequence as hilbert function and has the weak lefschetz property furthermore we show that g has maximal graded betti numbers among all arithmetically gorenstein subschemes of projective space whose artinian reduction has the weak lefschetz property and the given hilbert function as an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h vector contains a polytope with maximal graded betti numbers

closure solvability for network coding and secret sharing network coding is a new technique to transmit data through a network by letting the intermediate nodes combine the packets they receive given a network the network coding solvability problem decides whether all the packets requested by the destinations can be transmitted in this paper we introduce a new approach to this problem we define a closure operator on a digraph closely related to the network coding instance and we show that the constraints for network coding can all be expressed according to that closure operator thus a solution for the network coding problem is equivalent to a so called solution of the closure operator we can then define the closure solvability problem in general which surprisingly reduces to finding secret sharing matroids when the closure operator is a matroid based on this reformulation we can easily prove that any multiple unicast where each node receives at least as many arcs as there are sources solvable by linear functions we also give an alternative proof that any nontrivial multiple unicast with two source receiver pairs is always solvable over all sufficiently large alphabets based on singular properties of the closure operator we are able to generalize the way in which networks can be split into two distinct parts we also provide a new way of identifying and removing useless nodes in a network we also introduce the concept of network sharing where one solvable network can be used to accommodate another solvable network coding instance finally the guessing graph approach to network coding solvability is generalized to any closure operator which yields bounds on the amount of information that can be transmitted through a network

gotzmann ideals of the polynomial ring let a k x 1 x n denote the polynomial ring in n variables over a field k we will classify all the gotzmann ideals of a with at most n generators in addition we will study hilbert functions h for which all homogeneous ideals of a with the hilbert function h have the same graded betti numbers these hilbert functions will be called inflexible hilbert functions we introduce the notion of segmentwise critical hilbert functions and show that segmentwise critical hilbert functions are inflexible

structural decomposition of monomial resolutions abstract we express the multigraded betti numbers of an arbitrary monomial ideal in terms of the multigraded betti numbers of two basic classes of ideals this decomposition has multiple applications in some concrete cases we use it to construct minimal resolutions of classes of monomial ideals in other cases we use it to compute projective dimensions to illustrate the effectiveness of the structural decomposition we give a new proof of a classic theorem by charalambous that states the following let k be a field and m an artinian monomial ideal in s k x 1 x n then for all i b i s m n i

non degenerate curves with maximal hartshorne rao module extending results for space curves we establish bounds for the cohomology of a non degenerate curve in projective n space as a consequence for any given n we determine all possible pairs d g where d is the degree and g is the arithmetic genus of the curve furthermore we show that curves attaining our bounds always exist and describe properties of these extremal curves in particular we determine the hartshorne rao module the generic initial ideal and the graded betti numbers of an extremal curve

graph theoretical constructions for graph entropy and network coding based communications the guessing number of a directed graph digraph equivalent to the entropy of that digraph was introduced as a direct criterion on the solvability of a network coding instance this paper makes two contributions on the guessing number first we introduce an undirected graph on all possible configurations of the digraph referred to as the guessing graph which encapsulates the essence of dependence amongst configurations we prove that the guessing number of a digraph is equal to the logarithm of the independence number of its guessing graph therefore network coding solvability is no more a problem on the operations made by each node but is simplified into a problem on the messages that can transit through the network by studying the guessing graph of a given digraph and how to combine digraphs or alphabets we are thus able to derive bounds on the guessing number of digraphs second we construct specific digraphs with high guessing numbers yielding network coding instances where a large amount of information can transit we first propose a construction of digraphs with finite parameters based on cyclic codes with guessing number equal to the degree of the generator polynomial we then construct an infinite class of digraphs with arbitrary girth for which the ratio between the linear guessing number and the number of vertices tends to one despite these digraphs being arbitrarily sparse these constructions yield solvable network coding instances with a relatively small number of intermediate nodes for which the node operations are known and linear although these instances are sparse and the sources are arbitrarily far from their corresponding sinks

generic initial ideals and squeezed spheres in 1988 kalai constructed a large class of simplicial spheres called squeezed spheres and in 1991 presented a conjecture about generic initial ideals of stanley reisner ideals of squeezed spheres in the present paper this conjecture will be proved in order to prove kalai s conjecture based on the fact that every squeezed d 1 sphere is the boundary of a certain d ball called a squeezed d ball generic initial ideals of stanley reisner ideals of squeezed balls will be determined in addition generic initial ideals of exterior face ideals of squeezed balls are determined on the other hand we study the squeezing operation which assigns to each gorenstein complex g having the weak lefschetz property a squeezed sphere sq g and show that this operation increases graded betti numbers

reverse lex ideals we study reverse lex ideals in a polynomial ring and compare their properties to those of lex ideals in particular we provide an analogue of green s theorem for reverse lex ideals we also compare the betti numbers of strongly stable and square free strongly stable monomial ideals to those of reverse lex ideals

free resolutions of lex ideals over a koszul toric ring in this paper we study the minimal free resolution of lex ideals over a koszul toric ring in particular we study in which toric ring r all lex ideals are componentwise linear we give a certain necessity and sufficiency condition for this property and show that lex ideals in a strongly koszul toric ring are componentwise linear in addition it is shown that in the toric ring arising from the segre product p p every hilbert function of a graded ideal is attained by a lex ideal and that lex ideals have the greatest graded betti numbers among all ideals having the same hilbert function

maximal families of gorenstein algebras the purpose of this paper is to study maximal irreducible families of gorenstein quotients of a polynomial ring r let gradalg h r be the scheme parametrizing graded quotients of r with hilbert function h we prove there is a close relationship between the irreducible components of gradalg h r whose general member is a gorenstein codimension c 1 quotient and the irreducible components of gradalg h r whose general member b is a codimension c cohen macaulay algebra of hilbert function h related to h if the castelnuovo mumford regularity of the gorenstein quotient is large compared to the castelnuovo mumford regularity of b this relationship actually determines a well defined injective mapping from such cohen macaulay components of gradalg h r to gorenstein components of gradalg h r in which generically smooth components correspond moreover the dimension of the gorenstein components is computed in terms of the dimension of the corresponding cohen macaulay component and a sum of two invariants of b using linkage by a complete intersection we show how to compute these invariants linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem

hilbert schemes and maximal betti numbers over veronese rings macaulay s theorem macaulay in proc lond math soc 26 531 555 1927 characterizes the hilbert functions of graded ideals in a polynomial ring over a field we characterize the hilbert functions of graded ideals in a veronese ring r the coordinate ring of a veronese embedding of pr 1 we also prove that the hilbert scheme which parametrizes all graded ideals in r with a fixed hilbert function is connected this is an analogue of hartshorne s theorem hartshorne in math ihes 29 5 48 1966 that hilbert schemes over a polynomial ring are connected furthermore we prove that each lex ideal in r has the greatest betti numbers among all graded ideals with the same hilbert function

non linear behaviour of castelnuovo mumford regularity abstract it is shown that the castelnuovo mumford regularity of lex segment ideals of ordinary powers as well as of frobenius powers of a homogeneous polynomial ideal is a function of polynomial type of huge degree

on the resolution of ideals of fat points abstract in this paper we determine the hilbert function and the minimal system of generators of r 1 n 1 general fat points of p n furthermore by looking at the minimal system of generators we can show that the ideal associated to two fat points in p n or the ideal associated to r 1 n general fat points all with the same multiplicities is a splittable ideal and this is the first step in constructing a minimal resolution

generic sequences of polynomials abstract let i be an ideal in a polynomial ring s over an infinite field such that i is generated by a generic sequence of homogeneous polynomials of specified degrees froberg has conjectured a formula for the hilbert series of s i moreno socias has conjectured a combinatorial property for the initial ideal of i with respect to degree reverse lexicographic order i show that moreno socias conjecture implies froberg s conjecture i also give a criterion for a hilbert series to admit an ideal with the property proposed by moreno socias and show that the hilbert series proposed by froberg does have this property

betti numbers of chordal graphs and f vectors of simplicial complexes abstract let g be a chordal graph and i g its edge ideal let i g 0 1 p denote the betti sequence of i g where i stands for the ith total betti number of i g and where p is the projective dimension of i g it will be shown that there exists a simplicial complex of dimension p whose f vector f f 0 f 1 f p coincides with i g

bounds for betti numbers abstract in this paper we prove parts of a conjecture of herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded kth syzygy module over the polynomial ring if in addition the module is z n graded we show that the conjecture holds in full generality furthermore we give lower and upper bounds for the graded betti numbers of graded ideals with a linear resolution and a fixed number of generators

fixed point theorems for boolean networks expressed in terms of forbidden subnetworks abstract we are interested in fixed points in boolean networks i e functions f from 0 1 n to itself we define the subnetworks of f as the restrictions of f to the subcubes of 0 1 n and we characterize a class f of boolean networks satisfying the following property every subnetwork of f has a unique fixed point if and only if f has no subnetwork in f this characterization generalizes the fixed point theorem of shih and dong which asserts that if for every x in 0 1 n there is no directed cycle in the directed graph whose the adjacency matrix is the discrete jacobian matrix of f evaluated at point x then f has a unique fixed point then denoting by c resp c the networks whose interaction graph is a positive resp negative cycle we show that the non expansive networks of f are exactly the networks of c c and for the class of non expansive networks we get a dichotomization of the previous forbidden subnetwork theorem every subnetwork of f has at most resp at least one fixed point if and only if f has no subnetworks in c resp c subnetwork finally we prove that if f is a conjunctive network then every subnetwork of f has at most one fixed point if and only if f has no subnetworks in c

resolutions of a stable ideals abstract let a a 1 a 2 a n be a sequence of integers or we introduce a stable ideals in a polynomial ring and study their homological properties our results generalize results on square free monomial ideals by aramova avramov herzog hibi and srinivasan

the weak and strong lefschetz properties for artinian k algebras abstract let a i 0ai be a standard graded artinian k algebra where chark 0 then a has the weak lefschetz property if there is an element l of degree 1 such that the multiplication l a i a i 1 has maximal rank for every i and a has the strong lefschetz property if l d a i a i d has maximal rank for every i and d the main results obtained in this paper are the following 1 every height three complete intersection has the weak lefschetz property our method surprisingly uses rank two vector bundles on p 2 and the grauert mulich theorem 2 we give a complete characterization including a concrete construction of the hilbert functions that can occur for k algebras with the weak or strong lefschetz property and the characterization is the same one 3 we give a sharp bound on the graded betti numbers achieved by our construction of artinian k algebras with the weak or strong lefschetz property and fixed hilbert function this bound is again the same for both properties some hilbert functions in fact force the algebra to have the maximal betti numbers 4 every artinian ideal in k x y possesses the strong lefschetz property this is false in higher codimension

extremal point sets and gorenstein ideals the hilbert function of a homogeneous ideal in r k x0 xn k a field is a much studied object this is not surprising since the hilbert function encodes important algebraic combinatorial and geometric information about the ideal the fact that recent computer algebra developments have made the hilbert function computable has not only sustained interest in them but sparked interest in many new questions about them in this paper we will concentrate on the hilbert functions which are the hilbert functions of points in p from 11 we know that this is the same as studying 0 dimensional differentiable o sequences equivalently the hilbert functions of graded artinian quotients of k x1 xn in our earlier paper 9 we began a discussion of n type vectors and showed that they were in 1 1 correspondence with hilbert functions of doi 10 1006 aima 1998 1889 available online at http www idealibrary com on

the linear guessing number of undirected graphs abstract this paper investigates linear guessing number introduced by riis in connection with the network coding theory this paper establishes properties of linear guessing number on undirected graphs in particular lower and upper bounds for linear guessing number are established finally linear guessing numbers of product graphs are investigated

certain bibasic hypergeometric transformations and their applications abstract we have obtained two new transformation formulae with the help of bailey s transform one of which contains series involving two independent bases it has also been shown that some very interesting new multiple series identities of the rogers ramanujan type can be established

lefschetz properties and hyperplane arrangements abstract in this article we study the weak and strong lefschetz properties and the related notion of almost revlex ideal in the non artinian case proving that several results known in the artinian case hold also in this more general setting we then apply the obtained results to the study of the jacobian algebra of hyperplane arrangements

koszul homology and extremal properties of gin and lex for every homogeneous ideal i in a polynomial ring r and for every p dim r we consider the koszul homology h i p r i with respect to a sequence of p of generic linear forms the koszul betti number ijp r i is by definition the dimension of the degree j part of h i p r i in characteristic 0 we show that the koszul betti numbers of any ideal i are bounded above by those of the gin revlex gin i of i and also by those of the lex segment lex i of i we show that ijp r i ijp r gin i iff i is componentwise linear and that and ijp r i ijp r lex i iff i is gotzmann we also investigate the set gins i of all the gin of i and show that the koszul betti numbers of any ideal in gins i are bounded below by those of the gin revlex of i on the other hand we present examples showing that in general there is no j is gins i such that the koszul betti numbers of any ideal in gins i are bounded above by those of j

reduction and fixed points of boolean networks and linear network coding solvability linear network coding transmits data through networks by letting the intermediate nodes combine the messages they receive and forward the combinations toward their destinations the solvability problem asks whether the demands of all the destinations can be simultaneously satisfied by using linear network coding the guessing number approach converts this problem into determining the number of fixed points of coding functions f a n to a n over a finite alphabet a usually referred to as boolean networks if a 0 1 with a given interaction graph that describes which local functions depend on which variables in this paper we generalize the so called reduction of coding functions in order to eliminate variables we then determine the maximum number of fixed points of a fully reduced coding function whose interaction graph has a loop on every vertex since the reduction preserves the number of fixed points we then apply these ideas and results to obtain four main results on the linear network coding solvability problem first we prove that non decreasing coding functions cannot solve any more instances than routing already does second we show that the triangle free undirected graphs are linearly solvable if and only if they are solvable by routing this is the first classification result for the linear network coding solvability problem third we exhibit a new class of non linearly solvable graphs fourth we determine large classes of strictly linearly solvable graphs

on the index coding problem and its relation to network coding and matroid theory the index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad hoc networks an instance of the index coding problem includes a sender that holds a set of information messages x x1 xk and a set of receivers r each receiver x h in r needs to obtain a message x x and has prior side information consisting of a subset h of x the sender uses a noiseless communication channel to broadcast encoding of messages in x to all clients the objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers in this paper we analyze the relation between the index coding problem the more general network coding problem and the problem of finding a linear representation of a matroid in particular we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem specifically we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions

index coding with side information motivated by a problem of transmitting supplemental data over broadcast channels birk and kol infocom 1998 we study the following coding problem a sender communicates with n receivers r1 rn he holds an input x 0 01l n and wishes to broadcast a single message so that each receiver ri can recover the bit xi each ri has prior side information about x induced by a directed graph grain nodes ri knows the bits of a in the positions j i j is an edge of g g is known to the sender and to the receivers we call encoding schemes that achieve this goal indexcodes for 0 1 n with side information graph g in this paper we identify a measure on graphs the minrank which exactly characterizes the minimum length of linear and certain types of nonlinear index codes we show that for natural classes of side information graphs including directed acyclic graphs perfect graphs odd holes and odd anti holes minrank is the optimal length of arbitrary index codes for arbitrary index codes and arbitrary graphs we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph this bound holds even for randomized codes but has been shown not to be tight

on the lex plus powers conjecture abstract let s be a polynomial ring over a field and i s a homogeneous ideal containing a regular sequence of forms of degrees d 1 d c in this paper we prove the lex plus powers conjecture when the field has characteristic 0 for all regular sequences such that d i j 1 i 1 d j 1 1 for each i that is we show that the betti table of i is bounded above by the betti table of the lex plus powers ideal of i

linear normality of general linear sections and some graded betti numbers of 3 regular projective schemes in this paper we study graded betti numbers of any nondegenerate 3 regular algebraic set x in a projective space pn more concretely via generic initial ideals gins method we mainly consider tailing betti numbers whose homological index is at least codim x pn for this purpose we first introduce a key definition nd 1 property which provides a suitable ground where one can generalize the concepts such as being nondegenerate or of minimal degree from the case of varieties to the case of more general closed subschemes and give a clear interpretation on the tailing betti numbers next we recall basic notions and facts on gins theory and we analyze the generation structure of the reverse lexicographic rlex gins of 3 regular nd 1 subschemes as a result we present exact formulae for these tailing betti numbers which connect them with linear normality of general linear sections of x with a linear subspace of dimension at least codim x pn finally we consider some applications and related examples

on an extension of galligo s theorem concerning the borel fixed points on the hilbert scheme abstract given an ideal i and a weight vector w which partially orders monomials we can consider the initial ideal in w i which has the same hilbert function a well known construction carries this out via a one parameter subgroup of a gl n 1 which can then be viewed as a curve on the corresponding hilbert scheme galligo a galligo theoreme de division et stabilite en geometrie analytique locale ann inst fourier grenoble 29 2 1979 107 184 vii proved that if i is in generic coordinates and if w induces a monomial order up to a large enough degree then in w i is fixed by the action of the borel subgroup of upper triangular matrices we prove that the direction the path approaches this borel fixed point on the hilbert scheme is also borel fixed

dynamics of positive automata networks with an important class of automata networks we associate a lyapunov function and by doing so we characterize its dynamic behaviour transient and cycle lengths as particular cases we study threshold and majority networks

on identities of the rogers ramanujan type abstract an elementary approach to a number of identities of the rogers ramanujan type is given it is shown that analytic formulas like e g the rogers ramanujan the rogers selberg and the gollnitz gordon identities can be obtained essentially as consequences of the q binomial theorem and the q vandermonde formula

poset embeddings of hilbert functions and betti numbers abstract we study inequalities between graded betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps defined earlier by us in caviglia and kummini 2013 5 we show that if graded betti numbers do not decrease when we replace ideals in an algebra by their embedded versions then the same behaviour is carried over to ring extensions as a corollary we give alternative inductive proofs of earlier results of bigatti hulett pardue mermin peeva stillman and murai we extend a hypersurface restriction theorem of herzog and popescu to the situation of embeddings we show that we can obtain the betti table of an ideal in the extension ring from the betti table of its embedded version by a sequence of consecutive cancellations we further show that the lex plus powers conjecture of evans reduces to the artinian situation