on the noetherianity of some associative finitely presented algebras abstract we consider finitely generated associative algebras over a fixed field k of arbitrary characteristic for such an algebra a we impose some structural restrictions we call a strictly ordered we are interested in the implication of strict order on a for its noetherian properties in particular we prove that if a is a graded standard finitely presented strictly ordered algebra then a is left noetherian if and only if it is almost commutative in this case a has polynomial growth

lattice polytopes cut out by root systems and the koszul property we show that lattice polytopes cut out by root systems of classical type are normal and koszul generalizing a well known result of bruns gubeladze and trung in type a we prove similar results for cayley sums of collections of polytopes whose minkowski sums are cut out by root systems the proofs are based on a combinatorial characterization of diagonally split toric varieties

rationality for generic toric rings we study generic toric rings we prove that they are golod rings so the poincare series of the residue field is rational we classify when such a ring is koszul and compute its rate also resolutions related to the initial ideal of the toric ideal with respect to reverse lexicographic order are described

multigraded castelnuovo mumford regularity we develop a multigraded variant of castelnuovo mumford regularity motivated by toric geometry we work with modules over a polynomial ring graded by a finitely generated abelian group as in the standard graded case our definition of multigraded regularity involves the vanishing of graded components of local cohomology we establish the key properties of regularity its connection with the minimal generators of a module and its behavior in exact sequences for an ideal sheaf on a simplicial toric variety x we prove that its multigraded regularity bounds the equations that cut out the associated subvariety we also provide a criterion for testing if an ample line bundle on x gives a projectively normal embedding

invertible completions of band matrices abstract starting with a hermitian band matrix r we fill it in as we try to complete it to an invertible hermitian matrix f often we can make f invertible even though all but one of the maximal hermitian submatrices in the band may be not invertible this paper is a continuation of the author s recent paper on bordered matrices

the period index problem for the brauer group of an algebraic surface in this paper we show that the period equals the index for elements of brauer groups of function fields of surfaces a key idea of the proof is that any azumaya algebra over a surface can be transformed into an azumaya algebra that is unobstructed

the ring of evenly weighted points on the line let m w mathbb p 1 n hbox sl 2 denote the geometric invariant theory quotient of mathbb p 1 n by the diagonal action of hbox sl 2 using the line bundle mathcal o w 1 w 2 ldots w n on mathbb p 1 n let r w be the coordinate ring of m w we give a closed formula for the hilbert function of r w which allows us to compute the degree of m w the graded parts of r w are certain kostka numbers so this hilbert function computes stretched kostka numbers if all the weights w i are even we find a presentation of r w so that the ideal i w of this presentation has a quadratic grobner basis in particular r w is koszul we obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of m w

multigraded regularity and the koszul property abstract we give a criterion for the section ring of an ample line bundle to be koszul in terms of multigraded regularity we discuss applications to adjoint bundles on toric varieties as well as to polytopal semigroup rings

frobenius splitting of schubert varieties and linear syzygies introduction let g be a connected simply connected semi simple alge braic group let b be a borel subgroup of g for an ample line bundle l on g b the questions about projective normality of the embedding of g b in the complete linear system of l and the equations defining the embedding were extensively studied ramanathan in 13 proved that the ideal defining the embedding of g b in the complete linear system of l is generated by quadratic equations kempf and ramanathan proved that the equations defining the embedding of g b in the complete linear system of l lr are quadratic and that a schubert variety x is linearly defined with respect to l lr ramanathan had conjectured that the method of frobenius splitting used in 13 can be used to study higher syzygies let r bh g b ljj where the vector mj ranges in nr be the multigraded coordinate ring of g b with respect to l lr in this paper we use frobenius splittings to study the higher syzygies of the irrelevant maximal ideal m of r we prove that m as a module has linear syzygies our methods also give simpler proofs of the results obtained in 6 and 13 let z be a projective variety defined over an algebraically closed field let l l be ample line bundles on z and let m denote the irrelevant maximal ideal of the multi graded ring r h z ljj in the first section we give a criterion for m to have linear syzigies over r we show that if for every n 2 the first cohomology of the n fold product zn with values in certain ideal sheaves tensored with powers of lj vanishes then m has linear syzygies propositions 1 9 and 1 10 ye also give a similar criterion for a homogeneous quotient ring of r to have linear syzygies over r proposition 1 13 in the second section we show that any r ample line bundles l l on g b and their restrictions to a schubert variety x c g b over an algebracally closed field of positive characteristic satisfy the above criteria to prove this result we use the method of frobenius splittings if z is a frobenius split variety and if l is an ample line bundle on z then we have hl z l 0 for 1 further if y is a compatibly split subvariety of z then the restriction

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

on the inertias of symmetric matrices and bounded self adjoint operators abstract let s be a hermitian matrix and s 1 a principal submatrix with r less rows and columns let and 1 be the numbers of positive eigenvalues of s and s 1 one of poincare s inequalities says that 1 r 1 we tighten these inequalities as follows let and 1 be dimkers and dimkers 1 and let d dimkers kers 1 then 1 r d 1 1 d

fibers of flat morphisms and weierstrass preparation theorem abstract we characterize flat extensions of commutative rings satisfying the weierstrass preparation theorem using this characterization we prove a variant of the weierstrass preparation theorem for rings of functions on a normal curve over a complete local domain of dimension one this generalizes recent works of harbater hartmann and krashen with a different method of proof

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

convergent presentations and polygraphic resolutions of associative algebras several constructive homological methods based on noncommutative grobner bases are known to compute free resolutions of associative algebras in particular these methods relate the koszul property for an associative algebra to the existence of a quadratic grobner basis of its ideal of relations in this article using a higher dimensional rewriting theory approach we give several improvements of these methods we define polygraphs for associative algebras as higher dimensional linear rewriting systems that generalise the notion of noncommutative grobner bases and allow more possibilities of termination orders than those associated to monomial orders we introduce polygraphic resolutions of associative algebras giving a categorical description of higher dimensional syzygies for presentations of algebras we show how to compute polygraphic resolutions starting from a convergent presentation and how these resolutions can be linked with the koszul property

normality of torus orbit closures in g p abstract the purpose of this note is to classify the torus orbit closures in an arbitrary algebraic homogeneous space g p that are toric varieties

geometry of pure spinor formalism in superstring theory abstract projective embedding of an isotropic grassmannian ogr 5 10 into projective space of spinor representation s can be characterized with a help of matrices by equations i 0 a polynomial function of degree n with values in s defines a map to ogr 5 10 if its coefficients satisfy a 2 n 1 quadratic equations algebra generated by coefficients of such polynomials is a coordinate ring of the quantum isotropic grassmannian we show that this ring is based on a lattice its defining relations satisfy straightened law this enables us to compute the poincare series of the ring

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

computer assistance for discovering formulas in system engineering and operator theory abstract the objective of this paper is two fold first we present a methodology for using a combination of computer assistance and human intervention to discover highly algebraic theorems in operator matrix and linear systems engineering theory since the methodology allows limited human intervention it is slightly less rigid than an algorithm we call it a strategy the second objective is to illustrate the methodology by deriving four theorems the presentation of the methodology is carried out in three steps the first step is introducing an abstraction of the methodology which we call an idealized strategy this abstraction facilitates a high level discussion of the ideas involved idealized strategies cannot be implemented on a computer the second and third steps introduce approximations of these abstractions which we call prestrategy and strategy respectively a strategy is more general than a prestrategy and in fact every prestrategy is a strategy the above mentioned approximations are implemented on a computer we stress that since there is a computer implementation the reader can use these techniques to attack their own algebra problems thus the paper might be of both practical and theoretical interest to analysts engineers and algebraists now we give the idea of a prestrategy a prestrategy relies almost entirely on two commands which we call ncprocess1 and ncprocess2 these two commands are sufficiently powerful so that in many cases when one applies them repeatedly to a complicated collection of equations they transform the collection of equations into an equivalent but substantially simpler collection of equations a loose description of a prestrategy applied to a list of equations is emsp 1 declare which variables are known and which are unknown at the beginning of a prestrategy the order in which the equations are listed is not important since ncprocess1 and ncprocess2 will reorder them so that the simplest ones appear first 2 apply ncprocess1 to the equations the output is a set of equations usually some in fewer unknowns than before carefully partitioned based upon which unknowns they contain 3 the user must select important equations especially any which solve for an unknown sayx when an equation is declared to be important or a variable is switched from being an unknown to being a known then the way in which ncprocess1 and ncprocess2 reorder the equations is modified 4 switchxto being known rather than unknown go to 2 above or stop when this procedure stops it hopefully gives the canonical necessary conditions for the original equations to have a solution as a final step we run ncprocess2 which aggressively eliminates redundant equations and partitions the output equations in a way which facilitates proving that the necessary conditions are also sufficient many classical theorems in analysis can be viewed in terms of solving a collection of equations we have found that this procedure actually discovers the classic theorem in a modest collection of classic cases involving factorization of engineering systems and matrix completion problems one might regard the question of which classical theorems in analysis can be proven with a strategy as an analog of classical euclidean geometry where a major question was what can be constructed with a compass and ruler here the goal is to determine which theorems in systems and operator theory could be discovered by repeatedly applying ncprocess1 and ncprocess2 or their successors and the human selection of equations which are important the major practical challenge addressed here is finding operations which when implemented in software present the user with crucial algebraic information about his problem while not overwhelming him with too much redundant information this paper consists of two parts a description of strategies a high level description of the algorithms a description of the applications to operator matrix and linear system engineering theory and a description of how one would use a strategy to discover four different theorems are presented in the first part of the paper thus one who seeks a conventional viewpoint for this rather unconventional paper might think of this as providing a unified proof of four different theorems the theorems were selected for their diverse proofs and because they are widely known so that many readers should be familiar with at least one of them the ncprocess commands use noncommutative grobner basis algorithms which have emerged in the last decade together with algorithms for removing redundant equations and a method for assisting a mathematician in writing a noncommutative polynomial as a composition of polynomials the reader needs to knownothingabout grobner basis to understand the first part of this paper descriptions involving the theory of grobner basis appear in the second part of the paper

diagonal subalgebras of bigraded algebras and embeddings of blow ups of projective spaces let v be closed subscheme of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i defined by a homogeneous ideal i a k x 1 x n and let x be the n 1 fold obtained by blowing up inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i along v if one embeds x in some projective space one is led to consider the subalgebra k i e c of a for some positive integers c and e the aim of this paper is to study ring theoretic properties of k i e c this is achieved by developing a theory which enables us to describe the local cohomology of certain modules over generalized segre products of bigraded algebras these results are applied to the study of the cohen macaulay property of the homogeneous coordinate ring of the blow up of the projective space along a complete intersection we also study the koszul property of diagonal subalgebras of bigraded standard k algebras

convergent relaxations of polynomial optimization problems with noncommuting variables we consider optimization problems with polynomial inequality constraints in noncommuting variables these noncommuting variables are viewed as bounded operators on a hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints such problems arise naturally in quantum theory and quantum information science to solve them we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution we also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem

regularity of modules over a koszul algebra abstract we prove that every module over a commutative homogeneous koszul algebra has regularity bounded by its regularity over a polynomial ring of which the koszul algebra is a homomorphic image from this we derive a result conjectured by george kempf to the effect that a sufficiently high truncation of any module over a homogeneous koszul algebra has a linear free resolution

subalgebras of bigraded koszul algebras abstract we show that diagonal subalgebras and generalized veronese subrings of a bigraded koszul algebra are koszul we give upper bounds for the regularity of side diagonal and relative veronese modules and apply the results to symmetric algebras and rees rings

noetherian properties of skew polynomial rings with binomial relations in this work we study standard finitely presented associative algebras over a fixed field k a restricted class of skew polynomial rings with quadratic relations considered in an earlier work of m artin and w schelter will be studied we call them binomial skew polynomial algebras we establish necessary and sufficient conditions for such an algebra to be a noetherian domain

an introduction to commutative and noncommutative gro bner bases abstract in 1965 buchberger introduced the notion of grobner bases for a polynomial ideal and an algorithm buchberger algorithm for their computation since the end of the seventies grobner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in algebraic geometry and commutative algebra moreover grobner bases have been also generalized to free noncommutative algebra and to various noncommutative algebras of interest in differential algebra e g weyl algebras enveloping algebras of lie algebras the aim of this paper is to give an introduction as elementary as i was able to make it to both commutative and noncommutative algebras grobner bases are in a sense a finite model of an infinite linear gauss reduced basis of an ideal viewed as a vector space and buchberger algorithm is the corresponding generalization of the gaussian elimination algorithm moreover the paper contains a survey of some applications of buchberger theory to noncommutative algebras together with these results surveyed this paper contains some minor new points e g the useless pair criteria in the noncommutative case and the final result on the existence and computability of grobner bases for two sided ideals in any finitely presented algebra

reduction of permutation invariant polynomials let r be a commutative ring with 1 let r x1 xn i be the polynomial algebra in the n 4 noncommuting variables x1 xn over r modulo the set of commutator relations i x1 xn xi xi x1 xn 1 i n furthermore let g be an arbitrary group of permutations operating on the indeterminates x1 xn and let r x1 xn ig be the r algebra of g invariant polynomials in r x1 xn i the first part of this paper is about an algorithm which computes a representation for any f r x1 xn ig as a polynomial in multilinear g invariant polynomials i e the maximal variable degree of the generators of r x1 xn ig is at most 1 the algorithm works for any ring r and for any permutation group g in addition we present a bound for the number of necessary generators for the representation of all g invariant polynomials in r x1 xn ig with a total degree of at most d the second part contains a first but promising analysis of g invariant polynomials of solvable polynomial rings

on syzygies of abelian varieties in this paper we prove the following result let x be a co mplex torus and m a normally generated line bundle on x then for every p 0 the line bundle mp 1 satisfies property np of green lazarsfeld

groups and lie algebras corresponding to the yang baxter equations for a positive integer n we introduce quadratic lie algebras tr n qtr n and finitely discrete groups tr n qtr n naturally associated with the classical and quantum yang baxter equation respectively we prove that the universal enveloping algebras of the lie algebras tr n qtr n are koszul and compute their hilbert series we also compute the cohomology rings for these lie algebras which by koszulity are the quadratic duals of the enveloping algebras finally we construct a basis of u tr n we construct cell complexes which are classifying spaces of the groups tr n and qtr n and show that the boundary maps in them are zero which allows us to compute the integral cohomology of these groups we show that the lie algebras tr n qtr n map onto the associated graded algebras of the malcev lie algebras of the groups tr n qtr n respectively in the case of tr n we use quantization theory of lie bialgebras to show that this map is actually an isomorphism at the same time we show that the groups tr n and qtr n are not formal for n 4

elimination of variables in linear solvable polynomial algebras and holonomicity abstract let k be a field of characteristic 0 based on the gelfand kirillov dimension computation of modules over solvable polynomial k algebras where solvable polynomial algebras are in the sense of a kandri rody and v weispfenning 1990 j symbolic comput 9 1 26 we prove that the elimination lemma obtained from d zeilberger 1990 j comput appl math 32 321 368 by using holonomic modules over the weyl algebra a n k and used in the automatic proving of special function identities holds for a class of solvable polynomial k algebras without any holonomicity restriction this opens a way to the solution of the extension contraction problem stemming from the automatic proving of multivariate identities with respect to the finiteness in the sense of f chyzak and b salvy 1998 j symbolic comput 26 187 227 it also yields a holonomicity so that automatic proving of multivariate identities may be dealt with by manipulating polynomial function coefficients instead of rational functions

one sided noncommutative grobner bases with applications to computing green s relations abstract standard noncommutative grobner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields this paper describes grobner basis procedures for one sided ideals in finitely presented noncommutative algebras over fields the polynomials defining a k algebra a as a quotient of a free k algebra are combined with the polynomials defining a one sided ideal i of a by using a tagging notation whose essential effect is to forbid left multiplication standard noncommutative grobner basis techniques can then be applied to the mixed set of polynomials thus calculating a i whilst working in a free structure avoiding the complication of computing in a the paper concludes by showing how the results can be applied to completable presentations of semigroups and so enable calculations of green s relations

koszul incidence algebras affine semigroups and stanley reisner ideals abstract we prove a theorem unifying three results from combinatorial homological and commutative algebra characterizing the koszul property for incidence algebras of posets and affine semigroup rings and characterizing linear resolutions of squarefree monomial ideals the characterization in the graded setting is via the cohen macaulay property of certain posets or simplicial complexes and in the more general nongraded setting via the sequential cohen macaulay property

test for leibniz n algebra structure abstract for a given basis of a vector space l over a field k and a multiplication table which is defined by a multilinear map l n l we present an algorithm and develop a computer program on mathematica in order to test if the given multiplication table corresponds to a lie n algebra or a non lie leibniz n algebra or neither the algorithm is based on the computation of a grobner basis of the ideal which is used in the construction of the universal enveloping algebra of the leibniz algebra provided by the daletskii takhtajan functor by means of the ncalgebra package which permits the construction of grobner bases in noncommutative associative algebras

koszul configurations of points in projective spaces abstract we prove a new criterion for the homogeneous coordinate ring of a finite set of points in p n to be koszul like the well known criterion due to kempf g kempf syzygies for points in projective space j algebra 145 1992 219 223 it involves only incidence conditions on linear spans of subsets of a given set we also give a sufficient condition for the koszul property to be preserved when passing to a subset of a finite set of points in p n

smooth points of t stable varieties in g b and the peterson map let g be a connected semi simple algebraic group defined over an algebraically closed field k and let t b p be respectively a maximal torus a borel subgroup and a parabolic subgroup of g inspired by a beautiful result of dale peterson describing the singular locus of a schubert variety in g b we characterize the t fixed points in the singular locus of an arbitrary irreducible t stable subvariety of g p a t variety for short peterson s result cf the deformation theorem 1 says that if k then a schubert variety x g b is smooth at a t fixed point x if and only if it is smooth at every t fixed point y x in the bruhat chevalley order on the fixed point set x t and all the limits c x x lim z x t z x z c c t of the zariski tangent spaces t z x of x coincide as c varies over the set of all t stable curves in x with c t x y where y x using this peterson showed that if g is simply laced and defined over then every rationally smooth point of a schubert variety in g b is smooth more generally the deformation c x x is defined for any k variety x with a t action provided c is what we call good i e c is a curve of the form c overline tz where z is a smooth point of x x t and x c t our first main result theorem 1 4 says that if x x is an attractive fixed point then x is smooth at x if and only if there exist at least two good c containing x such that c x x te x x where te x x denotes the span of the tangent lines of the t stable curves in x containing x in addition if x is cohen macaulay at x and c x x te x x for even one good c then x is smooth at x our second main result theorem 1 6 says that if x is a t variety in g p where g is simply laced then c x x te x x for each good c this is not true for general g but when g has no g2 factors then c x x is always contained in the linear span of the reduced tangent cone to x at x these results lead to several descriptions of the smooth fixed points of a t variety in g p and in particular they give simple proofs of peterson s results valid for any algebraically closed field we also show cf example 7 1 that there can exist t stable subvarieties in g b where g is simply laced which have rationally smooth t fixed points in their singular loci

normal presentation on elliptic ruled surfaces from the introduction let x be an irreducible projective variety and l a very ample lline bundle on x whose complete linear series defines l x p h0 l let s 1 m 0 smh0 x l and let r l l1 n 0 h0 x l n be the homogeneous coordinate ring associated to l then r is a finitely generated graded module over s so it has a minimal graded free resolution we say that the line bundle l is normally generated if the natural maps smh0 x l h0 x l m are surjective for all m 2 if l is normally generated then we say that l satisfies property np if the matrices in the free resolution of r over s have linear entries until the p th stage in particular property n1 says that the homogeneous ideal i of x in p h0 l is generated by quadrics a line bundle satisfying property n1 is also called normally presented let r kr1r2 be a graded algebra over a field k the algebra r is a koszul ring iff torri k k has pure degree i for all i in this article we determine exactly theorem 4 2 which line bundles on an elliptic ruled surface x are normally presented as a corollary we show that mukai s conjecture is true for the normal presentation of the adjoint linear series for an elliptic ruled surface in section 5 of this article we show that if l is normally presented on x then the homogeneous coordinate ring associated to l is koszul we also give a new proof of the following result due to butler if deg l 2g 2 on a curve x of genus g then l embeds x with koszul homogeneous coordinate ring

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

matroids and geometric invariant theory of torus actions on flag spaces abstract let f t be a geometric invariant theory quotient of a partial flag variety f sl n c p by the action t g p t g p of the maximal torus t in sl n c where p is a parabolic subgroup containing t the construction of f t depends upon the choice of a t linearized line bundle l of f this note concerns the case l l is a very ample homogeneous line bundle determined by a dominant weight meaning the associated character t c extends to p and to no larger parabolic subgroup if v denotes the irreducible representation of sl n c with highest weight and v is the isotypic component corresponding to a weight of the torus then f t is equal to proj n 0 v n n the weight is used to twist the canonical t linearization of l where the canonical t linearization of l is obtained by restricting the unique sl n c linearization of l to t we apply a theorem of gel fand goresky macpherson and serganova concerning matroid polytopes to show that if v 0 then one gets a well defined map f t cp dim v 1 by taking any basis of v equivalently all the semistable partial flags are detected by degree one t invariants provided v is nonzero we also show that the closure of any t orbit in f is projectively normal for the projective embedding f cp dim v 1

a polyhedral method for solving sparse polynomial systems a continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations we introduce mixed subdivisions of newton polytopes and we apply them to give a new proof and algorithm for bernstein s theorem on the expected number of roots this results in a numerical homotopy with the optimal number of paths to be followed in this homotopy there is one starting system for each cell of the mixed subdivision and the roots of these starting systems are obtained by an easy combinatorial construction

convex normality of rational polytopes with long edges abstract we introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths guaranteeing the property as an application we show that if every edge of a lattice d polytope p has lattice length 4 d d 1 then p is normal this answers in the positive a question raised in 2007 if p is a lattice simplex whose edges have lattice lengths d d 1 then p is even covered by lattice parallelepipeds for the approach developed here it is necessary to involve rational polytopes even for the application to lattice polytopes

weierstrass preparation and algebraic invariants we prove a form of the weierstrass preparation theorem for normal algebraic curves over complete discrete valuation rings while the more traditional algebraic form of weierstrass preparation applies just to the projective line over a base our version allows more general curves this result is then used to obtain applications concerning the values of u invariants and on the period index problem for division algebras over fraction fields of complete two dimensional rings our approach uses patching methods and matrix factorization results that can be viewed as analogs of cartan s lemma

local galois theory in dimension two this paper proves a generalization of shafarevich s conjecture for fields of laurent series in two variables over an arbitrary field this result says that the absolute galois group gk of such a field k is quasi free of rank equal to the cardinality of k i e every non trivial finite split embedding problem for gk has exactly cardk proper solutions we also strengthen a result of pop and haran jarden on the existence of proper regular solutions to split embedding problems for curves over large fields our strengthening concerns integral models of curves which are two dimensional

local global principles for torsors over arithmetic curves we consider local global principles for torsors under linear algebraic groups over function fields of curves over complete discretely valued fields the obstruction to such a principle is a version of the tate shafarevich group and for groups with rational components we compute it explicitly and show that it is finite this yields necessary and sufficient conditions for local global principles to hold our results rely on first obtaining a mayer vietoris sequence for galois cohomology and then showing that torsors can be patched we also give new applications to quadratic forms and central simple algebras

regularity and k0 group of quadric solvable polynomial algebras abstract concerning solvable polynomial algebras in the sense of kandri rody and weispfenning j symbolic comput 9 1990 1 26 it is shown how to recognize and construct quadric solvable polynomial algebras in an algorithmic way if a k a1 an is a quadric solvable polynomial algebra it is proved that gl dima n and k 0 a z if a is a tame quadric solvable polynomial algebra it is shown that a is completely constructable and auslander regular

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

on the shellability of the order complex of the subgroup lattice of a finite group we show that the order complex of the subgroup lattice of a finite group g is nonpure shellable if and only if g is solvable a by product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups

the structure of connected graded hopf algebras abstract in this paper we establish a structure theorem for connected graded hopf algebras over a field of characteristic 0 by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some desirable conditions the approach to the structure theorem is constructive based on the combinatorial properties of lyndon words and the standard bracketing on words as a surprising consequence of the structure theorem we show that connected graded hopf algebras of finite gelfand kirillov dimension over a field of characteristic 0 are all iterated hopf ore extensions of the base field in addition some keystone facts of connected hopf algebras over a field of characteristic 0 are observed as corollaries of the structure theorem without the assumptions of having finite gelfand kirillov dimension or affineness on hopf algebras or of that the base field is algebraically closed

the pythagoras number and the u invariant of laurent series fields in several variables abstract we show that every sum of squares in the three variable laurent series field r x y z is a sum of 4 squares as was conjectured in a paper of choi dai lam and reznick in the 1980 s we obtain this result by proving that every sum of squares in a finite extension of r x y is a sum of 3 squares it was already shown in choi dai lam and reznick s paper that every sum of squares in r x y itself is a sum of two squares we give a generalization of this result where r is replaced by an arbitrary real field our methods yield similar results about the u invariant of fields of the same type

the possible inertias for a hermitian matrix and its principal submatrices abstract definition a hermitian matrix h is a hermitian extension of a given set of hermitian matrices h ii i 1 m if these h ii are the block diagonals of h let i v i i in h ii the inertia of each h ii special case given hermitian matrices h ii i 1 m and given nonnegative integers v and such that v i v i i then a hermitian extension h exists such that ker h ker h ii and in h v if and only if i and max i and v max v i we also present a simple extension theorem for the general case ker h ker h ii

convergent arithmetic power series this paper considers rings of power series on discs of radius r 0 with coefficients lying in a subring a of the rational numbers one such ring denoted here by ar t is an arithmetic version of the disc algebra studied in analysis a slightly smaller ring ar t consists of the holomorphic functions on a neighborhood of the closed disc of radius r with coefficients lying in a the rings z1 t and z1 t are in fact equal to z t and so the families zr ti 0 r i and zr t i 0 r 1 can be regarded as interpolating between z t and z t these convergent power series rings may thus be of use in a deformation theoretic context the maximal spectra of ar t and ar t are described in section 1 and the latter ring is shown to be a noetherian regular unique factorization domain the ring zr t meanwhile is shown to be complete under the uniform norm section 2 shows that every finitely generated projective module over ar t is free the proof uses a patching lemma for free modules this section also contains a more general module patching result which can be viewed as an arithmetic analog of grothendieck s existence theorem from this point of view z is analogous to the polynomial ring k xl over a field k z t and z t correspond to k x t and k x t and zr t corresponds to k t x whose spectrum is a tubular neighborhood of the affine k line section 3 considers the subring ar kitih of ar t consisting of algebraic power series every such power series is shown to lie in an etale neighborhood of the disc it is also shown that the ring z t h of algebraic power series over z on the open unit disc consists entirely of rational functions given an algebraic structure over z t h obtained for example by

topological rewriting systems applied to standard bases and syntactic algebras abstract we introduce topological rewriting systems as a generalisation of abstract rewriting systems where we replace the set of terms by a topological space abstract rewriting systems correspond to topological rewriting systems for the discrete topology we introduce the topological confluence property as an approximation of the confluence property using a representation of linear topological rewriting systems with continuous reduction operators we show that the topological confluence property is characterised by lattice operations using this characterisation we show that standard bases induce topologically confluent rewriting systems on formal power series finally we investigate duality for reduction operators that we relate to series representations and syntactic algebras in particular we use duality for proving that an algebra is syntactic or not

shellability is np complete we prove that for every d 2 deciding if a pure d dimensional simplicial complex is shellable is np hard hence np complete this resolves a question raised e g by danaraj and klee in 1978 our reduction also yields that for every d 2 and k 0 deciding if a pure d dimensional simplicial complex is k decomposable is np hard for d 3 both problems remain np hard when restricted to contractible pure d dimensional complexes another simple corollary of our result is that it is np hard to decide whether a given poset is cl shellable