the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

linear algebraic methods in communication complexity abstract the notion of communication complexity seeks to capture the amount of communication between different parties that is required to find the output of a boolean function when each party is provided with only part of the input different variants of the model governing the rules of this communication lead to different connections with problems in combinatorial linear algebra in particular problems arise in this context that concern the rank of a 0 1 matrix and the minimum rank of a matrix meeting a given combinatorial description this paper surveys these connections

graphs whose positive semi definite matrices have nullity at most two abstract let g v e be a undirected graph containing n vertices and let m g be the set of all hermitian n n matrices m m i j with m i j 0 if i and j are connected by one edge of g with m i j c if i and j are connected by at least two edges with m i j 0 if i j and i and j are not connected by an edge of g and with m i i for i 1 n a real number what is the largest nullity attained by any positive semi definite matrix m m g in this paper we characterize for t 1 and 2 those graphs g for which the maximum nullity is not greater than t

communication complexity of simultaneous messages in the multiparty communication game cfl game of chandra furst and lipton proceedings of the 15th annual acm symposium on theory of computing boston ma 1983 pp 94 99 k players collaboratively evaluate a function f x0 xk 1 in which player i knows all inputs except xi the players have unlimited computational power the objective is to minimize communication in this paper we study the simultaneous messages sm model of multiparty communication complexity the sm model is a restricted version of the cfl game in which the players are not allowed to communicate with each other instead each of the k players simultaneously sends a message to a referee who sees none of the inputs the referee then announces the function value we prove lower and upper bounds on the sm complexity of several classes of explicit functions our lower bounds extend to randomized sm complexity via an entropy argument a lemma establishing a tradeoff between average hamming distance and range size for transformations of the boolean cube might be of independent interest our lower bounds on sm complexity imply an exponential gap between the sm model and the cfl model for up to log n 1 epsilon players for any epsilon 0 this separation is obtained by comparing the respective complexities of the generalized addressing function gafg k where g is a group of order n we also combine our lower bounds on sm complexity with the ideas of hastad and goldmann comput complexity 1 1991 pp 113 129 to derive superpolynomial lower bounds for certain depth 2 circuits computing a function related to the gaf function we prove some counterintuitive upper bounds on sm complexity we show that sf gaf mathbb z 2 t 3 has sm complexity o n 0 92 when the number of players is at least c log n for some constant c 0 our sm protocol for sf gaf mathbb z 2 t k has polylog n complexity we also examine a class of functions defined by certain depth 2 circuits this class includes the generalized inner product function and majority of majorities when the number of players is at least 2 log n we obtain polylog n upper bounds for this class of functions

locality for quantum systems on graphs depends on the number field adapting a definition of aaronson and ambainis 2005 theory comput 1 47 79 we call a quantum dynamics on a digraph saturated z local if the nonzero transition amplitudes specifying the unitary evolution are in exact correspondence with the directed edges including loops of the digraph this idea appears recurrently in a variety of contexts including angular momentum quantum chaos and combinatorial matrix theory complete characterization of the digraph properties that allow such a process to exist is a long standing open question that can also be formulated in terms of minimum rank problems we prove that saturated z local dynamics involving complex amplitudes occur on a proper superset of the digraphs that allow restriction to the real numbers or even further the rationals consequently among these fields complex numbers guarantee the largest possible choice of topologies supporting a discrete quantum evolution a similar construction separates complex numbers from the skew field of quaternions the result proposes a concrete ground for distinguishing between complex and quaternionic quantum mechanics

rounds in communication complexity revisited the k round two party communication complexity was studied in the deterministic model by p h papadimitriou and m sipser proc of the 14th stoc 1982 pp 330 337 and p duris z galil and g schnitger proc of the 16th stoc 1984 pp 81 91 and in the probabilistic model by a c yao proc of the 24th focs 1983 pp 420 428 and b halstenberg and r reischuk proc of the 20th stoc 1988 pp 162 172 this paper presents new lower bounds that give 1 randomization is more powerful than determinism in k round protocols and 2 an explicit function which exhibits an exponential gap between its k and k 1 round randomized complexity this paper also studies the three party communication model and exhibits an exponential gap in 3 round protocols that differ in the starting player finally this paper shows new connections of these questions to circuit complexity that motivate further work in this direction

unitary matrix digraphs and minimum semidefinite rank for an undirected simple graph g the minimum rank among all positive semidefinite matrices with graph g is called the minimum semidefinite rank msr of g in this paper we show that the msr of a given graph may be determined from the msr of a related bipartite graph finding the msr of a given bipartite graph is then shown to be equivalent to determining which digraphs encode the zero nonzero pattern of a unitary matrix we provide an algorithm to construct unitary matrices with a certain pattern and use previous results to give a lower bound for the msr of certain bipartite graphs

generalized tjurina numbers of an isolated complete intersection singularity we define a series of non negative integer valued invariants open image in new window the generalized tjurina numbers of an analytic n dimensional isolated complete intersection singularity the tjurina number equals 1 we show that all these numbers are bounded by the milnor number by establishing explicit formulas for the differences p involving the mixed hodge numbers of the cohomology of the link and a new range of analytic invariants for x x this generalizes earlier results of looijenga and steenbrink on the relation between the milnor number and the tjurina number we show that singularities satisfying 1 n 1 behave cohomologically like weighted homogeneous singularities and we expect this to be a criterion for weighted homogeneity

linearly independent vertices and minimum semidefinite rank we study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs we present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph and conjecture that the resulting graph invariant is equal to minimum semidefinite rank rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs we also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex

the minrank of random graphs the minrank of a directed graph g is the minimum rank of a matrix m that can be obtained from the adjacency matrix of g by switching some ones to zeros i e deleting edges and then setting all diagonal entries to one this quantity is closely related to the fundamental information theoretic problems of linear index coding bar yossef et al network coding effros et al and distributed storage mazumdar isit 2014 we prove tight bounds on the minrank of directed erd s renyi random graphs g n p for all regimes of p in 0 1 in particular for any constant p we show that mathsf minrk g theta n log n with high probability where g is chosen from g n p this bound gives a near quadratic improvement over the previous best lower bound of omega sqrt n haviv and langberg and partially settles an open problem raised by lubetzky and stav our lower bound matches the well known upper bound obtained by the clique covering solution and settles the linear index coding problem for random knowledge graphs

milnor number and tjurina number of complete intersections two basic invariants of a complex analytic isolated complete intersection singularity x x are its milnor number g x x and its tjurina number z x x the former is the n th betti number minus one if n 0 of a no nsingular nearby fibre whereas the latter is the dimension of the base space of a miniversal deformation of x x it is conjectured that for n 1 we always have p x x z x x so far this has been verified in the following cases 1 if x x is a hypersurface singularity 2 if n 1 3 if the link of x in x is a rational homology sphere 4 if x x is quasi homogeneous then we actually have equality of p x x and z x x 5 if n 2 the first case is rather trivial the next three are all due to greuel 3 and the fifth case is due to the first author 4 subsequently j wahl expressed for n 2 the difference of milnor number and tjurina number as a sum of non negative nvariants of the germ x x 7 the purpose of this note is to prove the conjecture n general in fact we obtain a higher dimensional generalization of wahl s result

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

writing representations over proper division subrings abstract let e be a division ring and g a finite group of automorphisms of e whose elements are distinct modulo inner automorphisms of e let f e g be the division subring of elements of e fixed by g given a representation a e d d of an f algebra a we give necessary and sufficient conditions for to be writable over f here e d d denotes the algebra of d d matrices over e and a matrix a writes over f if a 1 a a f d d we give an algorithm for constructing an a or proving that no a exists the case of particular interest to us is when e is a field and is absolutely irreducible the algorithm relies on an explicit formula for a and a generalization of hilbert s theorem 90 that arises in galois cohomology the algorithm has applications to the construction of absolutely irreducible group representations especially for solvable groups and to the recognition of class c 5 in aschbacher s matrix group classification scheme m aschbacher on the maximal subgroups of the finite classical groups invent math 76 1984 469 514 mr0746539 shangzhi li on the subgroup structure of classical groups in group theory in china in math appl china ser vol 365 kluwer acad publ dordrecht 1996 pp 70 90 mr1447199 1 13

on solving relative norm equations in algebraic number fields let q f be algebraic number fields and m f a free o e module we prove a theorem which enables us to determine whether a given relative norm equation of the form n f e has any solutions m at all and if so to compute a complete set of nonassociate solutions finally we formulate an algorithm using this theorem consider its algebraic complexity and give some examples

nonnegative ranks decompositions and factorizations of nonnegative matrices the nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank one matrices into which the matrix can be decomposed additively such decompositions are useful in diverse scientific disciplines we obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm

matrix rank and communication complexity abstract the rank of a matrix seems to play a role in the context of communication complexity a framework developed to analyze basic communication requirements of computational problems we present some issues and open problems arising in this area and put forward a number of research subjects in linear algebra whose investigation would shed new lights into the intriguing relationship between communication complexity and matrix rank we also mention the related problem of the accuracy of bounds on the chromatic number of a graph given in terms of the rank of its adjacency matrix

boolean circuits tensor ranks and communication complexity we investigate two methods for proving lower bounds on the size of small depth circuits namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices our methods are combinatorial but we think that our main contribution concerns the algebraic concepts used in this area tensor ranks and rigidity our main results are following i an o n bit protocol for a communication game for computing shifts which also gives an upper bound of o n 2 on the contact rank of the tensor of multiplication of polynomials this disproves some earlier conjectures a related probabilistic construction gives an o n upper bound for computing all permutations and an o n log log n upper bound on the communication complexity of pointer jumping with permutations ii a lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of logarithmic depth circuits this bound has interpretations both as a lower bound on the rigidity of the tensor of multiplication of polynomials and as a lower bound on the communication needed to compute the shift function in a restricted model iii an upper bound on boolean circuits of depth 2 for computing shifts and more generally all permutations this shows that such circuits are more efficient than the model based on sending bits along vertex disjoint paths

orthogonal representations minimum rank and graph complements orthogonal representations are used to show that complements of certain sparse graphs have positive semidefinite minimum rank at most 4 this bound applies to the complement of a 2 tree and to the complement of a unicyclic graph hence for such graphs the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph the minimum rank of the complement of a 2 tree is determined exactly

a and the vanishing topology of discriminants suppose thatf n 0 p 0 is finitely a determined withn p we define a milnor fiber for the discriminant off it is the discriminant of a stabilization off we prove that this discriminant milnor fiber has the homotopy type of a wedge of spheres of dimensionp 1 whose number we denote by f one of the main theorems of the paper is a type result if n p is in the range of nice dimensions in the sense of mather then mu delta f geqq a e codium with equality iff is weighted homogeneous outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization these results are further extended to nonlinear sections of free divisors

algebraic complexities and algebraic curves over finite fields abstract we consider the problem of minimal multiplicative complexity of polynomial multiplication and multiplication in finite extensions of fields for infinite fields minimal complexities are known winograd s 1977 math syst theory 10 169 180 we prove lower and upper bounds on minimal complexities over finite fields both linear in the number of inputs using the relationship with linear coding theory and algebraic curves over finite fields

the minimum semidefinite rank of a triangle free graph abstract we employ a result of moshe rosenfeld to show that the minimum semidefinite rank of a triangle free graph with no isolated vertex must be at least half the number of its vertices we define a rosenfeld graph to be such a graph that achieves equality in this bound and we explore the structure of these special graphs their structure turns out to be intimately connected with the zero nonzero patterns of the unitary matrices finally we suggest an exploration of the connection between the girth of a graph and its minimum semidefinite rank and provide a conjecture in this direction

zero forcing parameters and minimum rank problems the zero forcing number z g which is the minimum number of vertices in a zero forcing set of a graph g is used to study the maximum nullity minimum rank of the family of symmetric matrices described by g it is shown that for a connected graph of order at least two no vertex is in every zero forcing set the positive semidefinite zero forcing number z g is introduced and shown to be equal to g os g where os g is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank the positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs an example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex hermitian positive semidefinite minimum rank is presented

solving quadratic equations using reduced unimodular quadratic forms let q be an n x n symmetric matrix with integral entries and with det q 0 but not necesarily positive definite we describe a generalized lll algorithm to reduce this quadratic form this algorithm either reduces the quadratic form or stops with some isotropic vector it is proved to run in polynomial time we also describe an algorithm for the minimization of a ternary quadratic form when a quadratic equation q x y z 0 is solvable over q a solution can be deduced from another quadratic equation of determinant 1 the combination of these algorithms allows us to solve efficiently any general ternary quadratic equation over q and this gives a polynomial time algorithm as soon as the factorization of the determinant of q is known

solving norm equations in relative number fields using s units in this paper we are interested in solving the so called norm equation m l k x a where l k is a given arbitrary extension of number fields and a a given algebraic number of k by considering s units and relative class groups we show that if there exists at least one solution in l but not necessarily in z l then there exists a solution for which we can describe precisely its prime ideal factorization in fact we prove that under some explicit conditions the s units that are norms are norms of s units this allows us to limit the search for rational solutions to a finite number of tests and we give the corresponding algorithm when a is an algebraic integer we also study the existence of an integral solution and we can adapt the algorithm to this case

the sign rank of ac 0 the sign rank of a matrix a a ij with pm1 entries is the least rank of a real matrix b b ij with a ij b ij 0 for all i j we obtain the first exponential lower bound on the sign rank of a function in mathsf ac 0 namely let f x y bigwedge i 1 dots m bigvee j 1 dots m 2 x ij wedge y ij we show that the matrix f x y x y has sign rank exp omega m this in particular implies that sigma 2 cc not subseteq mathsf upp cc which solves a longstanding open problem in communication complexity posed by babai frankl and simon proceedings of the 27th symposium on foundations of computer science focs 1986 pp 337 347 our result additionally implies a lower bound in learning theory specifically let phi 1 dots phi r 0 1 n to mathbb r be functions such that every dnf formula f 0 1 n to 1 1 of polynomial size has the representation f equiv mathrm sgn a 1 phi 1 dots a r phi r for some reals a 1 dots a r we prove that then r geqslant exp omega n 1 3 which essentially matches an upper bound of exp tilde o n 1 3 due to klivans and servedio j comput system sci 68 2004 pp 303 318 finally our work yields the first exponential lower bound on the size of threshold of majority circuits computing a function in mathsf ac 0 this substantially generalizes and strengthens the results of krause and pudlak theoret comput sci 174 1997 pp 137 156