a short proof of a kupershmidt wilson theorem this is a short elementary proof of a statement originally observed by adler then pursued by the author by kupershmidt and wilson and in a more general setting by drinfeld and sokolov

bootstrapping variables in algebraic circuits we show that for the blackbox polynomial identity testing pit problem it suffices to study circuits that depend only on the first extremely few variables one needs only to consider size s degree s circuits that depend on the first log c s variables where c is a constant and composes a logarithm with itself c times thus the hitting set generator hsg manifests a bootstrapping behavior a partial hsg against very few variables can be efficiently grown to a complete hsg a boolean analog or a pseudorandom generator property of this type is unheard of our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w r t variables this is repeated c times in an efficient way pushing the envelope further we show that i a quadratic time blackbox pit for 6 913 variate degree s size s polynomials will lead to a near complete derandomization of pit and ii a blackbox pit for n variate degree s size s circuits in s n time for 1 2 will lead to a near complete derandomization of pit in contrast s n time is trivial our second idea is to study depth 4 circuits that depend on constantly many variables we show that a polynomial time computable o s 1 49 degree hsg for trivariate depth 4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of kabanets and impagliazzo kabanets v impagliazzo r 2003 proceedings of the thirty fifth annual acm symposium on theory of computing stoc 03

on zero testing and interpolation of k sparse multivariate polynomials over finite fields abstract given a black box which will produce the value of a k sparse multivariate polynomial for any given specific argument one may ask for optimal strategies 1 to distinguish such a polynomial from the zero polynomial 2 to distinguish any two such polynomials from one other and 3 to uniformly reconstruct the polynomial from such an information source while such strategies are known already for polynomials over fields of characteristic zero the equally important but considerably more complicated case of a finite field k of small characteristic is studied in the present paper the result is that the time complexity of such strategies depends critically on the degree m of the extension field of k from which the arguments are to be chosen e g if m equals the number n of variables then 1 can be solved by k 1 and 2 as well as 3 by 2 k 1 queries while in case m 1 essentially 2 log n log k queries are needed

extremal solutions of the two dimensionall problem of moments by using the theory of the principal function of a hyponormal operator with rank one self commutator one classifies some extremal points of the solutions of the truncatedl problem of moments in two real variables these extremal elements called degenerated solutions of thel problem are proved to be in a natural bijective correspondence with the bounded quadrature domains in the complex plane the main result of the paper characterizes these degenerated solutions from three distinct perspectives complex analysis moment problems and operator theory the interplay between the three points of view each with specific techniques constitutes the main body of the paper

the hamiltonian structures of the super kp hierarchy associated with an even parity super lax operator the even parity super lax operator is considered for the supersymmetric kp hierarchy of the form l d2 i 0 ui 2d i 1 and two hamiltonian structures are obtained following the standard method of gel fand and dikii it is observed that the first hamiltonian structure is local and linear whereas the second hamiltonian structure is nonlocal and nonlinear among the superfields appearing in the lax operator their connections with the super w algebra are discussed briefly

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

the hamiltonian structures associated with a generalized lax operator it is shown that with every lax operator which is a pseudodifferential operator of nonzero leading order is associated a kp hierarchy for each such operator we construct the second gelfand dikii bracket associated with the lax equation and show that it defines a hamiltonian structure when the leading order is positive the corresponding compatible first hamiltonian structure which turns out in general to be different from the naive first gelfand dikii bracket is derived the corresponding hamiltonian structures for the constrained lax operator where the next to leading order term vanishes or has a constant coefficient is discussed

the resultant on compact riemann surfaces we introduce a notion of the resultant of two meromorphic functions on a compact riemann surface and demonstrate its usefulness in several respects for example we exhibit several integral formulas for the resultant relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact riemann surface as a particular application the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the schottky double of the domain

selected topics on quadrature domains this is a selection of facts old and recent about quadrature domains the text written in the form of a survey is addressed to non experts and covers a variety of phenomena related to quadratur

remarkable matrices and trigonometric identities abstract two n n matrices are exhibited which have a simple expression in terms of trigonometric functions of n arbitrary angles and possess remarkably neat spectral properties such as integral eigenvalues several related trigonometric identities are also exhibited

integrable structure of the dirichlet boundary problem in two dimensions we study how the solution of the two dimensional dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem we show that the hadamard formula for the variation of the dirichlet green function under deformations of the domain reveals an integrable structure the independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain the solution to the dirichlet boundary problem is expressed through the tau function of the dispersionless toda hierarchy we also discuss a degenerate case of the dirichlet problem on the plane with a gap in this case the tau function is identical to the partition function of the planar large n limit of the hermitian one matrix model

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

ullemar s formula for the moment map ii we prove the complex analogue of ullemar s formula for the jacobian of the complex moment mapping this formula was previously established in the real case

faster integer multiplication for more than 35 years the fastest known method for integer multiplication has been the schonhage strassen algorithm running in time o n log n log log n under certain restrictive conditions there is a corresponding omega n log n lower bound all this time the prevailing conjecture has been that the complexity of an optimal integer multiplication algorithm is theta n log n we take a major step towards closing the gap between the upper bound and the conjectured lower bound by presenting an algorithm running in time n log n 2 o log n the running time bound holds for multitape turing machines the same bound is valid for the size of boolean circuits

commutative partial differential operators in one variable it is possible to describe explicitly the differential operators that commute with a given one at least when the centralizer of the given operator has rank 1 so far a generalization of the theory to several variables has been developed inexplicitly only for matrices whose size increases with the number of variables we propose to develop an algebraic theory of commuting partial differential operators pdos by formulating a generalization of the one variable techniques in particular darboux transformations and differential resultants in this paper we present a counter example to a one variable feature of maximal commutative rings and some facts examples and questions on differential resultants

algebraic independence and blackbox identity testing algebraic independence is a fundamental notion in commutative algebra that generalizes independence of linear polynomials polynomials f 1 f m k x 1 x n over a field k are called algebraically independent if there is no non zero polynomial f such that f f 1 f m 0 the transcendence degree trdeg f 1 f m is the maximal number r of algebraically independent polynomials in the set in this paper we design blackbox and efficient linear maps f that reduce the number of variables from n to r but maintain trdeg f f i i r assuming sparse f i and small r we apply these fundamental maps to solve two cases of blackbox identity testing assuming a large or zero characteristic 1 given a polynomial degree circuit c and sparse polynomials f 1 f m of transcendence degree r we can test blackbox d c f 1 f m for zeroness in poly size d r time 2 define a email protected email protected d k s n circuit to be of the form i 1 email protected j 1 sf i j where f i j are sparse n variate polynomials of degree at most d for this class of depth 4 circuits we define a notion of rank assuming there is a rank bound r for minimal simple email protected email protected d k s n identities we give a poly dsnr r k d 2 time blackbox identity test for email protected email protected d k s n circuits this partially generalizes the state of the art of depth 3 to depth 4 circuits the notion of transcendence degree works best with large or zero characteristic but we also give versions of our results for arbitrary fields

linear analysis of quadrature domains iii the linear data of a quadrature domain introduced in the first paper in this series is further investigated the matricial structure of all hyponormal operators with finite rank self commutators which is inductively constructed from such linear data is discussed in particular a conservation law of a determinantal function in the iterative process of defining these matrices is exploited the spectrum of the determining left upper corner of some of these operators is analyzed in connection with the existence of generalized quadrature formulas on planar domains

the zero curvature formulation of the skdv equations the fermionic extensions of the kdv equation are derived from the zero curvature condition associated with the superalgebra osp 2 1 this derivation clarifies why there are only two such extensions possible and why only one of them is supersymmetric a lenard type of derivation of these equations is also presented

hitting sets for roabp and sum of set multilinear circuits we give an n o log n time n is the input size blackbox polynomial identity testing algorithm for unknown order read once oblivious arithmetic branching programs roabps the best time complexity known for blackbox polynomial identity testing pit for this class was n o log 2 n due to forbes saptharishi and shpilka proceedings of the 2014 acm symposium on theory of computing 2014 pp 867 875 moreover their result holds only when the individual degree is small while we do not need any such assumption with this we match the time complexity for the unknown order roabp with the known order roabp due to forbes and shpilka proceedings of the 2013 ieee 54th annual symposium on foundations of computer science 2013 pp 243 252 and also with the depth 3 set multilinear circuits due to agrawal saha and saxena proceedings of the 2013 acm symposium on theory of computing 2013 pp 321 330 our proof is simpler and involves a new technique called basis isolation the depth 3

integrable structure of the dirichlet boundary problem in multiply connected domains we study the integrable structure of the dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply connected domains the solution to the dirichlet boundary problem in the multiply connected case is given through a quasiclassical tau function which generalizes the tau function of the dispersionless toda hierarchy it is shown to obey an infinite hierarchy of hirota like equations which directly follow from properties of the dirichlet green function and from the fay identities the relation to multi support solutions of matrix models is briefly discussed

on the bi hamiltonian structures of the skdv the first and the second hamiltonian structures of the supersymmetric kdv equation are derived following the conventional approach of gel fand and dickey in particular it is shown that the nonlocal nature of the first bracket is a direct consequence of the constrained nature of the lax operator

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces

multi component bi super hamiltonian kdv systems it is shown that a new class of classical multi component super kdv equations is bi super hamiltonian by extending the method for the verification of graded jacobi identity the multi component extension of super mkdv equations is obtained by using the super miura transformation