global holderian error bound for nondegenerate polynomials let f colon mathbb r n to mathbb r be a polynomial and s x in mathbb r n f x le 0 let f x max 0 f x if there exist c 0 alpha 0 beta 0 such that d x s le c f x alpha f x beta for all x in mathbb r n where d x s denotes the distance between x and s then we say that f has a global holderian error bound we prove that if f is convenient and nondegenerate with respect to its newton boundary at infinity in the sense of kouchnirenko invent math 32 1976 pp 1 32 then it has a global holderian error bound since the nondegenerate polynomials form a zariski open subset in a variety of polynomials with a given convenient newton boundary at infinity this result shows that global holderian error bounds hold for a large class of polynomials which can be relatively easy recognized

gauge transformation in einstein yang mills theories we discuss the relation between space time diffeomorphisms and gauge transformations in theories of the yang mills type coupled with einstein s general relativity we show that local symmetries of the hamiltonian and lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the legendre map although pure yang mills gauge transformations are projectable by themselves diffeomorphisms are not instead the projectable symmetry group arises from infinitesimal diffeomorphism inducing transformations which must depend on the lapse function and shift vector of the space time metric plus associated gauge transformations our results are generalizations of earlier results by ourselves and by salisbury and sundermeyer

equivalence classes of perturbations in cosmologies of bianchi types i and v interpretation in the preceding paper 26 gauge invariant variables were defined that characterize an almost bianchi type i or almost bianchi type v universe filled with a nonbarotropic perfect fluid one can think of these basic variables denoted collectively by d as having at least two aspects first d gives an explicit i e analytical representation of the equivalence class of perturbations in fact this equivalence class is uniquely determined from d and vice versa second any gauge invariant quantity with respect to a bianchi type i or bianchi type v background model is obtainable linearly from d through purely algebraic and differential operations among many other things the above properties of d facilitate new insights into the question of why a standard and well known formulation based on the stewart walker lemma does not enable one to describe the equivalence classes of perturbations and to find gauge invariant variables independent of d if we pose an analogous question in regard to almost robertson walker universe models a different but none the less instructive answer is obtained in this case the stewart walker lemma provides a complete framework both for constructing all gauge invariant variables and for determining the equivalence classes of perturbations because of the sometimes confusing statements in the literature nontrivial comparisons with other work on linear perturbations in anisotropic background models are made we also present what we believe are some interesting applications of our ideas to the subject of quantum field theory in curved space time

hoeffding decomposition in h 1 spaces the well known result of bourgain and kwapien states that the projection p le m onto the subspace of the hilbert space l 2 left omega infty right spanned by functions dependent on at most m variables is bounded in l p with norm le c p m for 1 p infty we will be concerned with two kinds of endpoint estimates we prove that p le m is bounded on the space h 1 left mathbb d infty right of functions in l 1 left mathbb t infty right analytic in each variable we also prove that p le 2 is bounded on the martingale hardy space associated with a natural double indexed filtration and more generally we exhibit a multiple indexed martingale hardy space which contains h 1 left mathbb d infty right as a subspace and p le m is bounded on it

surface charge algebra in gauge theories and thermodynamic integrability surface charges and their algebra in interacting lagrangian gauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus in the case of exact solutions and symmetries the surface charges are interpreted as a pfaff system integrability is governed by frobenius theorem and the charges associated with the derived symmetry algebra are shown to vanish in the asymptotic context we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended comparison with hamiltonian and covariant phase space methods is made all approaches are shown to agree for exact solutions and symmetries while there are differences in the asymptotic context

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

a global smooth version of the classical łojasiewicz inequality abstract let f r n r be a function of class c d d 1 such that d f x 1 d 0 on r n then the following global ojasiewicz inequality holds true c d dist x f 0 f x 1 0 d f x for all x r n where c d d 2 2 d 1 and dist x a denotes the euclidean distance from x to a as applications of this inequality we have the following statements if the sets f 0 and f x 1 0 are non asymptotic at infinity then there exist positive constants e and r such that c d dist x f 0 d f x whenever dist x f 0 e and x r if f is a polynomial of degree d with an isolated critical point at the origin the following effective ojasiewicz inequality holds true c dist x f 0 d 2 d 3 n 1 2 f x for all x r for some c 0 and r 0 finally we establish a relation between the above global ojasiewicz inequality and the phenomenon of singularities at infinity

lagrangian approach to the physical degree of freedom count in this paper we present a lagrangian method that allows the physical degree of freedom count for any lagrangian system without having to perform neither dirac nor covariant canonical analyses the essence of our method is to establish a map between the relevant lagrangian parameters of the current approach and the hamiltonian parameters that enter in the formula for the counting of the physical degrees of freedom as is given in dirac s method once the map is obtained the usual hamiltonian formula for the counting can be expressed in terms of lagrangian parameters only and therefore we can remain in the lagrangian side without having to go to the hamiltonian one using the map it is also possible to count the number of first and second class constraints within the lagrangian formalism only for the sake of completeness the geometric structure underlying the current approach developed for systems with a finite number of degrees of freedom is uncovered with the help of the covariant canonical formalism finally the method is illustrated in several examples including the relativistic free particle

covariant differential identities and conservation laws in metric torsion theories of gravitation i general consideration arbitrary diffeomorphically invariant metric torsion theories of gravity are considered it is assumed that lagrangians of such theories contain derivatives of field variables tensor densities of arbitrary ranks and weights up to a second order only the generalized klein noether methods for constructing manifestly covariant identities and conserved quantities are developed manifestly covariant expressions are constructed without including auxiliary structures like a background metric in the riemann cartan space the following manifestly generally covariant results are presented a the complete generalized system of differential identities the klein noether identities is obtained b the generalized currents of three types depending on an arbitrary vector field displacements are constructed they are the canonical noether current symmetrized belinfante current and identically conserved hilbert bergmann current in particular it is stated that the symmetrized belinfante current does not depend

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

division formulas on projective varieties we introduce a division formula on a possibly singular projective subvariety x of complex projective space mathbb p n which e g provides explicit representations of solutions to various polynomial division problems on the affine part of x especially we consider a global effective version of the briancon skoda huneke theorem

noether currents and charges for maxwell like lagrangians the hilbert noether theorem states that a current associated with diffeomorphism invariance of a lagrangian vanishes on shell modulo a divergence of an arbitrary superpotential application of the noether procedure to physical lagrangians yields however meaningful and measurable currents the well known solution to this paradox is to involve the variation of the metric tensor such procedure for the field considered on a fixed flat background is sophisticated logically one needs to introduce the variation of a fixed field and formal we analyse the noether procedure for a generic diffeomorphism invariant p form field model we show that the noether current of the field considered on a variable background coincides with the current treated in a fixed geometry consistent description of the canonical energy momentum current is possible only if the dynamics of the geometry gravitation is taken into account however even the truncated consideration yields the proper expression we examine the examples of the free p form gauge field theory the gr in the coframe representation and the metric free electrodynamics although the variation of a metric tensor is not acceptable in the latter case the noether procedure yields the proper result

general quadratic gauge theory constraint structure symmetries and physical functions how can we relate the constraint structure and constraint dynamics of the general gauge theory in the hamiltonian formulation to specific features of the theory in the lagrangian formulation especially relate the constraint structure to the gauge transformation structure of the lagrangian action how can we construct the general expression for the gauge charge if the constraint structure in the hamiltonian formulation is known whether we can identify the physical functions defined as commuting with first class constraints in the hamiltonian formulation and the physical functions defined as gauge invariant functions in the lagrangian formulation the aim of the present paper is to consider the general quadratic gauge theory and to answer the above questions for such a theory in terms of strict assertions to fulfil such a programme we demonstrate the existence of the so called superspecial phase space variables in terms of which the quadratic hamiltonian action takes a simple canonical form on the basis of such a representation we analyse a functional arbitrariness in the solutions of the equations of motion of the quadratic gauge theory and derive the general structure of symmetries by analysing a symmetry equation we then use these results to identify the two definitions of physical functions and thus prove the dirac conjecture

presymplectic current and the inverse problem of the calculus of variations the inverse problem of the calculus of variations asks whether a given system of partial differential equations pdes admits a variational formulation we show that the existence of a presymplectic form in the variational bicomplex when horizontally closed on solutions allows us to construct a variational formulation for a subsystem of the given pde no constraints on the differential order or number of dependent or independent variables are assumed the proof follows a recent observation of bridges hydon and lawson math proc cambridge philos soc 148 01 159 178 2010 and generalizes an older result of henneaux ann phys 140 1 45 64 1982 from ordinary differential equations odes to pdes uniqueness of the variational formulation is also discussed

global łojasiewicz type inequality for non degenerate polynomial maps abstract let f f 1 f p r n r p be a polynomial map this paper studies the existence of the following global ojasiewicz type inequality f x f x c d x f 1 0 for all x r n for some constants c 0 0 and 0 we show that the above inequality holds if one of the following conditions is satisfied i f is convenient and khovanskii non degenerate at infinity ii f is convenient and non degenerate at infinity iii f is mikhailov gindikin non degenerate further in cases ii and iii the exponents and can be determined explicitly

a global lojasiewicz inequality for algebraic varieties let x be the locus of common zeros of polynomials f 1 f in n complex variables a global upper bound for the distance to x is given in the form of a lojasiewicz inequality the exponent in this inequality is bounded by d min n k where d max 3 deg f i the estimates are also valid over an algebraically closed field of any characteristic

local symmetries and constraints the general relationship between local symmetries occurring in a lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied first a prescription applicable to an arbitrary lagrangian field theory for the construction of phase space from the manifold of field configurations on space time is given next a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses as special cases the usual gauge transformations of yang mills theory and general relativity local symmetries on phase space are then defined via projection from field configuration space it is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space moreover the constraints thereby obtained are always first class and the poisson bracket algebra of the constraint functions is isomorphic to the lie bracket algebra of the local symmetries on the constraint submanifold of phase space the differences that occur in the structure of constraints in yang mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space in yang mills theory all the field independent local symmetries project to all of phase space whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are field dependent as by products of the present work definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary lagrangian field theory and the noether current density associated with an arbitrary local symmetry a number of properties of these currents are established and some relationships between them are obtained

reduced phase space quotienting procedure for gauge theories we present a reduction procedure for gauge theories based on quotienting out the kernel of the presymplectic form in configuration velocity space local expressions for a basis of this kernel are obtained using phase space procedures the obstructions to the formulation of the dynamics in the reduced phase space are identified and circumvented we show that this reduction procedure is equivalent to the standard dirac method as long as the dirac conjecture holds that the dirac hamiltonian containing the primary first class constraints with their lagrange multipliers can be enlarged to an extended dirac hamiltonian which includes all first class constraints without any change of the dynamics the quotienting procedure is always equivalent to the extended dirac theory even when it differs from the standard dirac theory the differences occur when there are ineffective constraints and in these situations we conclude that the standard dirac method is preferable at least for classical theories an example is given to illustrate these features as well as the possibility of having phase space formulations with an odd number of physical degrees of freedom

covariant hamiltonian boundary conditions in general relativity for spatially bounded space time regions we investigate the covariant hamiltonian symplectic structure of general relativity for spatially bounded regions of space time with a fixed time flow vector for existence of a well defined hamiltonian variational principle taking into account a spatial boundary it is necessary to modify the standard arnowitt deser misner hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field the most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obtained from the symplectic current pulled back to the hypersurface boundary of the space time region a main result is that we obtain a covariant derivation of dirichlet neumann and mixed type boundary conditions on the gravitational field at a fixed boundary hypersurface together with the associated hamiltonian boundary terms as well we establish uniqueness of these boundary conditions under certain assumptions motivated by the form of the symplectic current our analysis uses a noether charge method which extends and unifies several results developed in recent literature for general relativity as an illustration of the method we apply it to the maxwell field equations to derive allowed boundary conditions and boundary terms for the existence of a well defined hamiltonian variational principle for an electromagnetic field in a fixed spatially bounded region of minkowski space time

łojasiewicz inequality at infinity for polynomials in two real variables we propose different types of ojasiewicz inequality at infinity for polynomials in two real variables the formulas for the ojasiewicz exponents are given

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem