sums of squares in real analytic rings let a be an analytic ring we show 1 a has finite pythagoras number if and only if its real dimension is 2 and 2 if every positive semidefinite element of a is a sum of squares then a is real and has real dimension 2

representations p adiques et equations differentielles in this paper we associate to every p adic representation v a p adic differential equation d rig v that is to say a module with a connection over the robba ring we do this via the theory of fontaine s k modules this construction enables us to relate the theory of k modules to p adic hodge theory we explain how to construct dcris v and dst v from d rig v which allows us to recognize semi stable or crystalline representations the connection is then unipotent or trivial on d rig v 1 t in general the connection has an infinite number of regular singularities but v is de rham if and only if those are apparent singularities a structure theorem for modules over the robba ring allows us to get rid of all singularities at once and to obtain a classical differential equation with a frobenius structure using this we construct a functor from the category of de rham representations to that of classical p adic differential equations with frobenius structure a recent theorem of y andre gives a complete description of the structure of the latter object this allows us to prove fontaine s p adic monodromy conjecture every de rham representation is potentially semi stable as an application we can extend to the case of arbitrary perfect residue fields some results of hyodo h1g h1st of perrin riou the semi stability of ordinary representations of colmez absolutely crystalline representations are of finite height and of bloch and kato if the weights of v are 2 then bloch kato s exponential expv is an isomorphism

some remarks on the computation of pythagoras numbers of real irreducible algebroid curves through gram matrices we clarify a difficulty that appears in r quarez j algebra 238 2001 139 to bound the pythagoras number of a real irreducible algebroid curve by its multiplicity

some remarks on pythagorean real curve germs let k be a real closed field a real ap curve over k is a 1 dimensional excellent henselian local real domain with residue field k a 1 dimensional noetherian local ring is arf if emb dim b mult b for every local ring b infinitely near to a j lipman amer j math 93 1971 649 685 for n 1 the 2nth pythagoras number p2n of a commutative ring a is the least p 1 p such that any sum of 2nth powers in a is a sum of no more than p2nth powers in a a main purpose of this paper is to affirm the following conjectures proposed by ruiz j algebra 94 1985 no 1 126 144 let a be a real ap curve and let a be pythagorean i e p2 1 then i a is arf ii every local ring infinitely near to a is pythagorean actually the authors obtain a finer result for a real ap curve a the following assertions are equivalent 1 a is arf 2 a is pythagorean 3 p2n 1 for some n 4 p2n 1 for all n here 2 1 is exactly conjecture i and 1 2 reduces conjecture ii to the obvious fact that if a is arf every local ring infinitely near to a is arf too of course the result contains some additional insight into the study of pythagoras s numbers even of higher order of real curve germs

decision trees old and new results abstract in this paper we prove two general lower bounds for algebraic decision trees which test membership in a set s r n which is defined by linear inequalities let rank s be the maximal dimension of a linear sub space contained in the closure of s in euclidean topology first we show that any decision tree for s which uses products of linear functions we call such functions mlf functions must have depth at least n rank s this solves an open question raised by a c yao and can be used to show that mlf functions are not really more powerful than simple comparisons between the input variables when computing the largest k out of n elements yao proved this result in the special case when products of at most two linear functions are allowed our proof also shows that any decision tree for this problem must have exponential size using the same methods we can give an alternative proof of rabin s theorem namely that the depth of any decision tree for s using arbitrary analytic functions is at least n rank s

sums of two squares in analytic rings we study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions this is a basic useful property of the plane but difficult to check in other cases in particular what about z 2 xy z 2 yx 2 y 3 z 2 x 3 y 4 or z 2 x 3 xy 3 in fact the unique positive examples we can find are the brieskorn singularity the union of two planes in 3 space and the whitney umbrella conversely we prove that a complete intersection with that property other than the seven embedded surfaces already mentioned must be a very simple deformation of the two latter namely in particular except for the stems z 2 x 2 and z 2 yx 2 all singularities are real rational double points

cellular automata as an alternative to rather than an approximation of differential equations in modeling physics abstract cellular automata are models of distributed dynamical systems whose structure is particularly well suited to ultrafast exact numerical simulation on the other hand they constitute a radical departure from the traditional partial differential equation approach to distributed dynamics here we discuss the problem of encoding the state variables and evolution laws of a physical system into this new setting and of giving suitable correspondence rules for interpreting the model s behavior

on the nullstellensatze for stein spaces and c analytic sets in this work we prove the real nullstellensatz for the ring o x of analytic functions on a c analytic set x rn in terms of the saturation of ojasiewicz s radical in o x the ideal i a of the zero set a of an ideal a of o x coincides with the saturation formula presented of ojasiewicz s radical formula presented if a has good properties concerning hilbert s 17th problem then i a formula presented where formula presented stands for the real radical of a the same holds if we replace formula presented with the real analytic radical formula presented of a which is a natural generalization of the real radical ideal in the c analytic setting we revisit the classical results concerning hilbert s nullstellensatz in the framework of complex stein spaces let a be a saturated ideal of o rn and yrn the germ of the support of the coherent sheaf that extends aorn to a suitable complex open neighborhood of rn we study the relationship between a normal primary decomposition of a and the decomposition of yrn as the union of its irreducible components if a p is prime then i p p if and only if the complex dimension of yrn coincides with the real dimension of p

a class of recursive permutations which is primitive recursive complete abstract we focus on total functions in the theory of reversible computational models we define a class of recursive permutations dubbed reversible primitive permutations rpp which are computable invertible total endo functions on integers so a subset of total reversible computations rpp is generated from five basic functions identity sign change successor predecessor swap two notions of composition sequential and parallel one functional iteration and one functional selection rpp is closed by inversion and it is expressive enough to encode cantor pairing and the whole class of primitive recursive functions

separation of semialgebraic sets we study the problem of deciding whether two disjoint semialgebraic sets of an algebraic variety over r are separable by a polynomial for that we isolate a dense subfamily of spaces of orderings named geometric which suffice to test separation and that reduce the problem to the study of the behaviour of the semialgebraic sets in their boundary then we derive several characterizations for the generic separation among which there is a geometric criterion that can be tested algorithmically finally we show how to check recursively whether we can pass from generic separation to separation obtaining a decision procedure for solving the problem

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

polynomial inequalities representing polyhedra our main result is that every n dimensional polytope can be described by at most 2n 1 polynomial inequalities and moreover these polynomials can explicitly be constructed for an n dimensional pointed polyhedral cone we prove the bound 2n 2 and for arbitrary polyhedra we get a constructible representation by 2n polynomial inequalities

analogue p adique du theoreme de turrittin et le theoreme de la monodromie p adique 1 rappel de quelques resultats sur la structure p adique d un point singulier d une equation differentielle 322 2 cas d une pente entiere 327 3 cas d une pente fractionnaire premiere a p 333 4 cas d un rang premier a p 335 5 cas general 339 6 l exemple mx 1 3 343 7 la theorie des modules et l implication dr pst 346

a convex optimization model for finding non negative polynomials this paper presents a convex optimization model for the problem of finding some polynomials for which certain linear combinations are non negative polynomials this model is then applied to solve several filter design problems we first reformulate some low pass filter design problems with finite or infinite impulse response as optimization problems over non negative real or complex polynomials whose feasibility problems can be solved by applying our model the whole optimization problems are then solved by using a combination of a bisection search procedure on an appropriate parameter and our convex optimization model to solve the feasibility problems some numerical examples illustrate the method

the length and other invariants of a real field the length of a field is the smallest integer m such that any totally positive quadratic form of dimension m represents all sums of squares we investigate this field invariant and compare it to others such as the u invariant the pythagoras number the hasse number and the mordell function related to sums of squares of linear forms

pythagoras numbers of real irreducible algebroid curves and gram matrices abstract using elementary techniques of linear algebra in relation with gram matrices we bound the pythagoras number of a real irreducible algebroid curve by its multiplicity

representing simple d dimensional polytopes by d polynomials a polynomial representation of a convex d polytope p is a finite set p 1 x p n x of polynomials over mathbb r d such that p x in mathbb r d p i x ge 0 mbox for every 1 le i le n let s d p be the least possible n as above it is conjectured that s d p d for all convex d polytopes p we confirm this conjecture for simple d polytopes by providing an explicit construction of d polynomials that represent a given simple d polytope p

radii of convergence and index for p adic differential operators we study the radii of p adic convergence of solutions at a generic point of homogeneous linear differential operators whose coefficients are analytic elements as an application we prove a conjecture of p robba for a certain class of operators concerning the relation between radii of convergence and index on analytic elements we also give an explicit factorization theorem for p adic differential operators based on the radii of generic convergence and the slopes of the associated newton polygon

rank one solvable p adic differential equations and finite abelian characters via lubin tate groups we introduce a new class of exponentials of artin hasse type called exponentials these exponentials depend on the choice of a generator of the tate module of a lubin tate group mathfrak g over mathbb z p they arise naturally as solutions of solvable differential modules over the robba ring if mathfrak g is isomorphic to widehat mathbb g m over mathbb z p we develop methods to test their over convergence and get in this way a stronger version of the frobenius structure theorem for differential equations we define a natural transformation of the artin schreier complex into the kummer complex this provides an explicit generator of the kummer unramified extension of epsilon dagger k infty whose residue field is a given artin schreier extension of k t where k is the residue field of k we then compute explicitly the group under tensor product of isomorphism classes of rank one solvable differential equations moreover we get a canonical way to compute the rank one module over epsilon dagger k infty attached to a rank one representation of mathrm gal k t mathrm sep k t defined by an artin schreier character

cohomological invariants for quadratic forms over local rings let a be local ring in which 2 is invertible and let n be a non negative integer we show that the nth cohomological invariant of quadratic forms is a well defined homomorphism from the nth power of the fundamental ideal in the witt ring of a to the degree n etale cohomology of a with mod 2 coefficients which is surjective and has kernel the hbox n 1 th power of the fundamental ideal this is obtained by proving the gersten conjecture for witt groups in an important mixed characteristic case

a p adic local monodromy theorem we produce a canonical filtration for locally free sheaves on an open p adic annulus equipped with a frobenius structure using this filtration we deduce a conjecture of crew on p adic differential equations analogous to grothendieck s local monodromy theorem also a consequence of results of andre and of mebkhout namely given a finite locally free sheaf on an open p adic annulus with a connection and a compatible frobenius structure the module admits a basis over a finite cover of the annulus on which the connection acts via a nilpotent matrix

on the pythagoras numbers of real analytic rings abstract we show that the pythagoras number of a real analytic ring of dimension 2 is finite bounded by a function of the multiplicity and the codimension

image directe superieure et unipotence let x be a proper smooth curve over a perfect field of characteristic p 0 and u an open dense subscheme of x we prove that convergent f isocrystals on u are overconvergent under the condition that they are overconvergent at each point in x nu using this criterion we show that the higher direct images rfcrys ov by a proper smooth morphism of schemes f v u are overconvergent introduction soient u k une courbe a ne et lisse sur un corps parfait de caracteristique p 0 x k une compactification lisse de u et f v u un morphisme propre et lisse en utilisant un critere de surconvergence dont l idee initiale revient a de jong cf 9 et gr ce aux resultats locaux de kedlaya 13 nous montrons que les images directes superieures rf ov sont surconvergentes repondant ainsi dans le cas d une courbe a ne et lisse a une conjecture de 3 en particulier leurs groupes de cohomologie sont de dimension finie et verifient la dualite de poincare enfin nous montrons que le theoreme de monodromie p adique local de 1 14 et 17 permet de deduire que tout f isocristal surconvergent sur u provient apres une eventuelle extension finie etale d un f log isocristal nous remercions kiran kedlaya pour nous avoir indique comment obtenir des resultats de descente de modules via ses lemmes de factorisations de matrices a notations soit k un corps parfait de caracteristique p on note w l anneau des vecteurs de witt de k ok une extension finie de w et k0 respectivement k le corps des fractions de w resp ok on suppose le corps k muni d un frobenius s relevant le frobenius canonique de k0 pour une indeterminee t on notera s 1 4 ok 1 21 2t nok k a 1 4 py i1 40 ait i ai a k jaijr i 0 i y pour tout 0 r 1 h 1 4 a 1 4 p y i1 40 ait i ai a k jaij 0 i y hy 1 4 a a h jaijr 0 i y pour un certain 0 r 1

on the approximation property of excellent rings where d is a smooth finite type a algebra a theorem which would imply the existence of an embedding of the above type yields the positive solution of the bass quillen conjecture so far this theorem is only known in special cases there is the following result by artin 3 4

on globally defined semianalytic sets in this work we present the concept of c semianalytic subset of a real analytic manifold and more generally of a real analytic space c semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by cartan c analytic sets for short more precisely s is a c semianalytic subset of a real analytic space x ox if each point of x has a neighborhood u such that s u is a finite boolean combinations of global analytic equalities and strict inequalities on x by means of paracompactness c emianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on x the family of c semianalytic sets is closed

sums of squares in real rings let a be an excellent ring we show that if the real dimension of a is at least three then a has in finite pythagoras number and there exists a positive semidefinite element in a which is not a sum of squares in a

positive semidefinite germs in real analytic surfaces we find all real analytic surface germs in mathbb r 3 on which every positive semidefinite function germ is a sum of squares in fact of two squares of analytic function germs

generation of invertible functions abstract we prove that all and only the invertible one variable partial recursive dunctions can be generated from six basic functions by means of functional composition and a new kind of functional iteration which may be called reversible iteration

overconvergent modular forms and the fontaine mazur conjecture we prove a conjecture of fontaine and mazur on modularity of representations of g which are potentially semi stable at p for representations coming from finite slope overconvergent eigenforms we also give an application of our technique to a question of gouvea on overconvergence of certain modular forms if is a suitable representation of g s on a two dimensional vector space over a finite field product of m and the s a finite set of primes we construct a rigid analytic subspace of the universal deformation space of which is defined by a purely representation theoretic condition and contains the eigencurve of coleman mazur conjecturally it is equal to this curve

semialgebraic topology over a real closed field i paths and components in the set of rational points of an algebraic variety we fix once and for all a real closed base field r by a variety x over r we mean a separated algebraic scheme x over r for all problems attacked here we could equally well assume that x is also reduced since we are only interested in the set x r of rational points of x notice that for every closed point x of x the residue class field k x ojmx either coincides with r i e x is rational or is isomorphic to r t we call the points x of x with k x r the real points of x and the other closed points the complex points of x r is a topological field a basis of open sets being given by the open intervals

uniform bounds on complexity and transfer of global properties of nash functions we show that the complexity of semialgebraic sets and mappings can be used to parametrize nash sets and mappings by nash families from this we deduce uniform bounds on the complexity of nash functions that lead to first order descriptions of many properties of nash functions and a good behaviour under real closed field extension e g primary decomposition as a distinguished application we derive the solution of the extension and global equations problems over arbitrary real closed fields in particular over the field of real algebraic numbers this last fact and a technique of change of base are used to prove that the artin mazur description holds for abstract nash functions on the real spectrum of any commutative ring and solve extension and global equations in that abstract setting to complete the view we prove the idempotency of the real spectrum and an abstract version of the separation problem we also discuss the conditions for the rings of abstract nash functions to be noetherian

on the pythagoras numbers of real analytic curves we show that the pythagoras number of a real analytic curve is the supremum of the pythagoras numbers of its singularities or that supremum plus 1 this includes cases when the pythagoras number is infinite

arithmetic and differential swan conductors of rank one representations with finite local monodromy we consider a complete discrete valuation field of characteristic p with possibly nonperfect residue field let rm v be a rank one continuous representation of its absolute galois group with finite local monodromy we will prove that the it arithmetic swan conductor of rm v defined after k kato which fits in the more general theory of abbes saito coincides with the it differential swan conductor of the associated differential module rm d dag rm v defined by k kedlaya this construction is a generalization to the nonperfect residue case of the fontaine s formalism as presented in the work of n tsuzuki our method of proof will allow us to give a new interpretation of the refined swan conductor

on the positive extension property and hilbert s 17th problem for real analytic sets in this work we study the positive extension pe property and hilbert s 17th problem for real analytic germs and sets a real analytic germ x 0 of r 0 n has the pe property if every positive semidefinite analytic function germ on x 0 has a positive semidefinite analytic extension to r 0 n analogously one states the pe property for a global real analytic set x in an open set q of r 0 n these pe properties are natural variations of hilbert s 17th problem here we prove that 1 a real analytic germ x 0 subset of r 0 3 has the pe property if and only if every positive semidefinite analytic function germ on x 0 is a sum of squares of analytic function germs on x 0 and 2 a global real analytic set x of dimension 2 and local embedding dimension 3 has the pe property if and only if it is coherent and all its germs have the pe property if that is the case every positive semidefinite analytic function on x is a sum of squares of analytic functions on x moreover we classify the singularities with the pe property

sums of squares in function fields over henselian local fields we give upper bounds for the level and the pythagoras number of function fields over fraction fields of integral henselian excellent local rings in particular we show that the pythagoras number of mathbb r x 1 dots x n is le 2 n 1 which answers positively a question of choi dai lam and reznick

sum of squares length of real forms for n d ge 1 let p n 2d denote the smallest number p such that every sum of squares of degree d forms in mathbb r x 1 ldots x n is a sum of p squares we establish lower bounds for p n 2d that are considerably stronger than the bounds known so far combined with known upper bounds they give p 3 2d in d 1 d 2 in the ternary case assuming a conjecture of iarrobino kanev on dimensions of tangent spaces to catalecticant varieties we show that p n 2d sim const cdot d n 1 2 for d rightarrow infty and all n ge 3 for ternary sextics and quaternary quartics we determine the exact value of the invariant showing p 3 6 4 and p 4 4 5

on the rationality and continuity of logarithmic growth filtration of solutions of p adic differential equations we study the asymptotic behavior of solutions of frobenius equations defined over the ring of overconvergent series as an application we prove chiarellotto tsuzuki s conjecture on the rationality and right continuity of dwork s logarithmic growth filtrations associated to ordinary linear p adic differential equations with frobenius structures

infinitesimal deformation of p adic differential equations on berkovich curves we show that if a differential equation mathscr f over a quasi smooth berkovich curve x has a certain compatibility condition with respect to an automorphism sigma of x then mathscr f acquires a semi linear action of sigma i e lifting that on x the compatibility condition forces the automorphism sigma to be close to the identity of x so the above construction applies to a certain class of automorphisms called infinitesimal this generalizes andre and di vizio asterisque 1 296 55 111 2004 and pulita compos math 144 4 867 919 2008 we also obtain an application to morita s p adic gamma function and to related values of p adic l functions

the convergence newton polygon of a p adic differential equation i affinoid domains of the berkovich affine line we prove that the radii of convergence of the solutions of a p adic differential equation fancyscript f over an affinoid domain x of the berkovich affine line are continuous functions on x that factorize through the retraction of x to gamma of x onto a finite graph gamma subseteq x we also prove their super harmonicity properties this finiteness result means that the behavior of the radii as functions on x is controlled by a finite family of data

sums of even powers in archimedean rings this note is devoted to the study of totally archimedean rings of regular functions we extend schmudgen s theorem to this class of rings moreover we show that in such rings every totally positive element is a sum of even powers of totally positive elements and hence is a sum of even powers of units

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

finite local monodromy of overconvergent unit root f isocrystals on a curve we prove that the category of p adic continuous representations of a fundamental group of a curve over a perfect field of a positive characteristic p with finite local monodromy is equivalent to that of overconvergent unit root f isocrystals on the curve to show the equivalence of categories we study the local theory and apply it to the global case

eigenvalues of symmetric matrices over integral domains abstract given an integral domain a we consider the set of all integral elements over a that can occur as an eigenvalue of a symmetric matrix over a we give a sufficient criterion for being such an element in the case where a is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element the latter result settles a questions raised by bass estes and guralnick

weighted sums of squares in local rings and their completions i let a be an excellent regular local ring of dimension two let t be a finitely generated preordering in a and let widehat t be the preordering generated by t in the completion widehat a of a we study the question when the property of being saturated descends from widehat t to t and establish conditions of geometric nature which allow to decide this question as an application we classify all principal preorderings in a of degree 3 which are saturated in the case where a has a real closed residue field these results have direct implications for nonnegativity certificates for real polynomials on two dimensional semi algebraic sets