Positivity, Valuations, Andquadratic Forms

To any number eld K and positive integer one can associate a constant which in the case ofK = Q coincides with the classical Hermite constant. Wewill survey some results concerning these constants, and how one can compute some of them for quadratic number elds. Hartwig Bosse (Frankfurt) Non-negative polynomials that are not sums of squares ursday, 11:00 Jaka Cimpric (Ljubljana) A method for computing lowest eigenvalues of symmetric polynomial dišerential operators by semidenite programming ursday, 17:15 Jeroen Demeyer (Ghent) Hilbert’s tenth problem for function elds over valued elds Friday, 10:45 Abstract. A diophantine equation is a polynomial equation with integer coe›cients in an arbitrary number of variables. Hilbert’s Tenth Problem was the following question: nd an algorithm which, given a diophantine equation, tells whether or not it has a solution over the integers. It was shown in 1970 by Y. Matiyasevich, building on earlier work by M. Davis, H. Putnam and J. Robinson, that this question has a negative answer: such an algorithm does not exist. In other words: diophantine equations are undecidable. Many authors have generalized this undecidability to other rings and elds. Two important open cases areQ andC(x). In this talk, I will show that equations overC((t))(x) are undecidable. is is a special case of a more general theorem which works for function elds over a valued eld in equal characteristic zero with some extra conditions. Quadratic forms play an important role in the proof. A diophantine equation is a polynomial equation with integer coe›cients in an arbitrary number of variables. Hilbert’s Tenth Problem was the following question: nd an algorithm which, given a diophantine equation, tells whether or not it has a solution over the integers. It was shown in 1970 by Y. Matiyasevich, building on earlier work by M. Davis, H. Putnam and J. Robinson, that this question has a negative answer: such an algorithm does not exist. In other words: diophantine equations are undecidable. Many authors have generalized this undecidability to other rings and elds. Two important open cases areQ andC(x). In this talk, I will show that equations overC((t))(x) are undecidable. is is a special case of a more general theorem which works for function elds over a valued eld in equal characteristic zero with some extra conditions. Quadratic forms play an important role in the proof. Max Dickmann (Paris) Faithfully quadratic rings Saturday, 12:25 Ronan Flatley (Dublin) Trace forms of symbol algebras Saturday, 10:35 Abstract. Let S be a symbol algebra. e trace form of S is computed and it is shown how this form can be used to determine whether S is a division algebra or not. Let S be a symbol algebra. e trace form of S is computed and it is shown how this form can be used to determine whether S is a division algebra or not. 4 POSITIVITY, VALUATIONS, AND QUADRATIC FORMS Wulf-Dieter Geyer (Erlangen) Patching Monday, 15:00 Abstract. In the last decades there have been developed at least four dišerent but related versions of patching: e analytic patching, the formal patching, the rigid patching and the algebraic patching. is talk specialises patching to the construction of galois covers, i.e. to the Inverse Galois Problem, especially in the analytic and in the algebraic case. In the last decades there have been developed at least four dišerent but related versions of patching: e analytic patching, the formal patching, the rigid patching and the algebraic patching. is talk specialises patching to the construction of galois covers, i.e. to the Inverse Galois Problem, especially in the analytic and in the algebraic case. Paveł Gładki (Santa Barabara) Certain quotient spaces of spaces of orderings and their inverse limits Tuesday, 11:30 Julia Hartmann (Aachen) Local global principles for quadratic forms Tuesday, 9:15 Abstract. is talk is concerned with local global principles for isotropy of quadratic forms over function elds of curves over complete discretely valued elds. We also obtain a short exact sequence for the corresponding Witt groups. e results are obtained using patching methods. (Joint work with David Harbater and Daniel Krashen) is talk is concerned with local global principles for isotropy of quadratic forms over function elds of curves over complete discretely valued elds. We also obtain a short exact sequence for the corresponding Witt groups. e results are obtained using patching methods. (Joint work with David Harbater and Daniel Krashen) Detlev Hošmann (Nottingham) Something old and something new about u ursday, 15:00 Abstract. e u-invariant resp. the Hasse number ũ of a eld is the smallest nonnegative integer n such that every torsion resp. totally indenite quadratic form of dimension > n is isotropic provided such an n exists, and it is dened to be innity otherwise. In the case of a nonreal eld, every form is a torsion form, and every form is totally indenite since this condition is vacuous in the absence of orderings, so both invariants coincide and are equal to the supremum of the dimensions of anisotropic forms over that eld. For real elds, one always has u ≤ ũ but equality need not hold. We present some old and new results on the relation between these invariants and on conditions that guarantee their niteness. e u-invariant resp. the Hasse number ũ of a eld is the smallest nonnegative integer n such that every torsion resp. totally indenite quadratic form of dimension > n is isotropic provided such an n exists, and it is dened to be innity otherwise. In the case of a nonreal eld, every form is a torsion form, and every form is totally indenite since this condition is vacuous in the absence of orderings, so both invariants coincide and are equal to the supremum of the dimensions of anisotropic forms over that eld. For real elds, one always has u ≤ ũ but equality need not hold. We present some old and new results on the relation between these invariants and on conditions that guarantee their niteness. Max Karoubi (Paris) Periodicity in hermitian K-theory Tuesday, 12:15 Abstract. It is known since few years, by the work of Voevodsky and Rost, that higher algebraic Ktheory of a commutative ring Awith suitable nite coe›cients is periodic above the étale dimension of A. In this lecture, we prove that any ring Awith periodic K-theory above a certain range has also a periodic K-theory for quadratic forms above this range. is lecture is prepared for a non specialist audience. erefore, all the basic denitions will be given. It is known since few years, by the work of Voevodsky and Rost, that higher algebraic Ktheory of a commutative ring Awith suitable nite coe›cients is periodic above the étale dimension of A. In this lecture, we prove that any ring Awith periodic K-theory above a certain range has also a periodic K-theory for quadratic forms above this range. is lecture is prepared for a non specialist audience. erefore, all the basic denitions will be given. Igor Klep (Ljubljana) Free real algebraic geometry Tuesday, 15:00 Abstract. In this talk we will sketch a few of the developments in the emerging area of real algebraic geometry over a free ∗-algebra, in particular on “noncommutative inequalities". Let n ∈ N, K ∈ {R,C} and letK⟨X , X∗⟩ denote the free ∗-algebra on X, that is, the set of allK-linear combinations of words in X ∶= (X , . . . , Xn) and X∗ ∶= (X∗  , . . . , X∗ n). Such elements are called NC polynomials. For f ∈ K⟨X , X∗⟩ and an n-tuple A = (A , . . . ,An) of matrices of the same size we consider the evaluation f (A) = f (A , . . . ,An ,A , . . . ,An). POSITIVITY, VALUATIONS, AND QUADRATIC FORMS 5 Two types of results will be discussed: one parallels classical real algebra with sums of squares representations (Positivstellensätze) for positive NC polynomials, and the other has a dišerent žavor focusing onNC semialgebraic sets andNCmappings between them. ese sets are dened as positivity domains of NC polynomials and include NC balls, and solution sets of NC linear matrix inequalities (LMIs). Here is a sample: eorem 1 (Helton). Suppose f ∈ K⟨X , X∗⟩ satises f (A) is positive semidenite for all n-tuples A of matrices of the same size. en f is a sum of hermitian squares. eorem 2 (Helton, McCullough, K.). Let h be an origin and boundary preserving NC mapping between NC balls Bn ∶= {B = (B , . . . , Bn) ∣ ∥B∥ ≤ } and Bm . en h is linear and there is a unique isometry U ∈ Km×n satisfying h = UX. In particular, if m < n then no such NC mapping exist. Manfred Knebusch (Regensburg) Semirings with bounds Monday, 10;45 Abtract. We call a (commutative) semiring R a semiring with upper bounds (or ub-semiring for short) if the addition on R gives a partial ordering on R such that, for any two elements x , y of R, the sum x + y is an upper bound of x and y. (It may be bigger than the maximum of x and y which perhaps does not exist). is new notion in semiring theory generalizes the notion of an upper bound group invented recently by Niels Schwartz. Ub-semirings give a natural frame to study families of valuations on semirings, and thus take a natural place in real algebra. e semiring of all sums of squares in a eld is a case in point. An important further class of ub-semirings are the supertropical semirings invented by Zur Izhakian. ey allow to rene the valuations on a semiring (in particular on a eld) to “supervaluations” (joint work with Zur Izhakian and Louis Rowen). Jochen Koenigsmann (Oxford) Fields with the Galois group of Q Friday, 9:15 Abstract. Using valuation theory and quadratic forms, we show that a eld F whose absolute Galois group GF = Gal(F se p/F) is isomorphic to GQ shares many arithmetic properties with Q. We will report on recent progress towards a classication of such elds and the impact on the birational Section Conjecture in Grothendieck’s Anabelian Geometry. Using valuation theory and quadratic forms, we show that a eld F whose absolute Galois group GF = Gal(F se p/F) is isomorphic to GQ shares many arithmetic properties with Q. We will report on recent progress towards a classication of such elds and the impact on the birational Section Conjecture in Grothendieck’s Anabelian Geometry. Monique Laurent (Amsterdam) Optimization over polynomials with sums of squares and moment matrices ursday, 9:30 Abstract. Polynomial optimization deals with the problem of minimizing a multivariate polynomial over a basic closed semi-algebraic set K dened by polynomial inequalities and equations. While polynomial time solvable when all polynomials are linear (via linear programming), the problem becomes hard in general as soon as it involves non-linear polynomials. Just adding the simple quadratic constraints x i = x i on the variables, already makes the problem NP-hard. A natural approach is then to relax the problem and to consider easier to solve, convex relaxations. e basic idea, which goes back to work of Hilbert, is to relax non-negative polynomials by sums of squares of polynomials, a notion which can be tested e›ciently (using semidenite programming algorithms). On the dual side, one views points as (atomic) measures and one searches for e›cient conditions characterizing sequences of moments of non-negativemeasures on the semi-algebraic set K. In this way hierarchies of e›cient convex relaxations can be build. We will review their various properties. In particular, convergence properties, that rely on real algebraic geometry representation results for positive polynomials, stopping criteria and extraction of global minimers, that rely on results from moment theory and commutative algebra. We will consider in particular unconstrained polynomial optimization problems and problems with a nite real variety. 6 POSITIVITY, VALUATIONS, AND QUADRATIC FORMS Noa Lavi (Be’er Sheva) Ganzstellensatz for open sets in real closed elds Monday, 11:30 Abstract. For valued elds, instead of being positive or non-negative, one might consider the property of being integral. A rational function will be called integral-denite over a denable set S if it takes only values in the valuation ring (ganze element). One can search for an algebraic representation of such functions. e aim of the talk will be showing a model theoretic framework for proving such ganzstellensatz theorem for open sets intersected by integrality set of n functions in real closed elds. Ha Nguyen (Atlanta) Polynomials non-negative on non-compact subsets of the plane Monday, 12:15 Abstract. Recently, M. Marshall answered a long-standing question by showing that if f (x , y) ∈ R[x , y] is non-negative on the strip [, ] ×R, then f has a representation f = σ + σ( − x), where σ , σ ∈ R[x , y] are sums of squares. In this talk we present some generalizations of this result to other non-compact basic closed semialgebraic sets of R which are contained in the strip. We also give some negative results. Recently, M. Marshall answered a long-standing question by showing that if f (x , y) ∈ R[x , y] is non-negative on the strip [, ] ×R, then f has a representation f = σ + σ( − x), where σ , σ ∈ R[x , y] are sums of squares. In this talk we present some generalizations of this result to other non-compact basic closed semialgebraic sets of R which are contained in the strip. We also give some negative results. Albrecht Pster (Mainz) An elementary and constructive proof of Hilbert’s theorm on ternary quartics Tuesday, 10:45 Janez Povh (Ljubljana) Ncsostools: a computer algebra system for symbolic and numerical computation with nc polynomials ursday, 12:20 Ronan Quarez (Rennes) Ešective tridiagonal determinantal representation for univariate polynomials and real roots counting Monday, 16:30 Abstract. We show how Sturm and Sylvester algorithms, which both compute the number of real roots of a given univariate polynomial over the reals, lead to two tridiagonal determinantal representations that can be viewed as dual. We show how Sturm and Sylvester algorithms, which both compute the number of real roots of a given univariate polynomial over the reals, lead to two tridiagonal determinantal representations that can be viewed as dual. Peter Roquette (Heidelberg) Arf invariants of quadratic forms in historical perspective Monday, 9:15 Abstract. Cahit Arf, a mathematician from Turkey, studied in 1937/38 as a doctoral student with Helmut Hasse. In his thesis he proved the Arf part of what now is known as the Hasse-Arf theorem which is of importance in class eld theory. During his stay in Göttingen he met Ernst Witt and there developed a friendship between the two, e topic of my lecture will be Arf ’s second paper where he transferredWitt’s theory of quadratic forms to elds of characteristic . In the course of his investigation he discovered what is now known as the Arf invariant of a quadratic form. is is of relevance to local and global elds of characteristic , and has also applications in topology. I will report on this from a historical perspective on the basis of letters, manuscripts and documents which are preserved at the HandschriŸenabteilung of the Göttingen library. ere are more than 60 letters between Hasse and Arf. In later years Cahit Arf has become a leading gure in the mathematics scene of Turkey. e new 10 Lira note of Turkish currency carries the portrait of Arf together with the formula: Arf(q) = ∑ i=. .n q(a i)q(b i) ∈ Z POSITIVITY, VALUATIONS, AND QUADRATIC FORMS 7 for Arf invariants. As I will point out this formula does not really represent the scope of Arf ”s discovery. Yuriy Savchuk (Leipzig) Positivstellensätze for some algebras of matrices Monday, 17:15 Konrad Schmüdgen (Leipzig) Positivity, sums of squares and positivstellensätze for ∗-algebras Friday, 16:30 Abstract. is talk will be concerned with various notions and results on positivity of symmetric elements of noncommutative ∗-algebras. Let A be a (complex or real) unital ∗-algebra with involution a → a∗ and letAh be the set of symmetric elements a = a∗ ofA. A subset C ofAh is called a quadratic module if  ∈ C, λc + λc ∈ C and a∗ca ∈ C for all c , c , c ∈ C , λ, λ ∈ [,∞) and a ∈ A. ere are quadratic modules dened in algebraic terms (as nite weigthed sums of squares a∗ca for all a ∈ A and c in some xed subset ofAh) and quadraticmodules dened byHilbert space representations (as those elements ofAh which are mapped into positive operators under some ∗-representations of A). Noncommutative Positivstellensätze deal with the interplay between these two classes of quadratic modules. Some new Positivstellensätze are presented for ∗-algebras of matrices over commutative or noncommutative ∗-algebras and for ∗-algebras of fractions. is talk will be concerned with various notions and results on positivity of symmetric elements of noncommutative ∗-algebras. Let A be a (complex or real) unital ∗-algebra with involution a → a∗ and letAh be the set of symmetric elements a = a∗ ofA. A subset C ofAh is called a quadratic module if  ∈ C, λc + λc ∈ C and a∗ca ∈ C for all c , c , c ∈ C , λ, λ ∈ [,∞) and a ∈ A. ere are quadratic modules dened in algebraic terms (as nite weigthed sums of squares a∗ca for all a ∈ A and c in some xed subset ofAh) and quadraticmodules dened byHilbert space representations (as those elements ofAh which are mapped into positive operators under some ∗-representations of A). Noncommutative Positivstellensätze deal with the interplay between these two classes of quadratic modules. Some new Positivstellensätze are presented for ∗-algebras of matrices over commutative or noncommutative ∗-algebras and for ∗-algebras of fractions. Andrew Smith (Konstanz) Fast positivity testing using Bernstein expansion ursday, 16:30 Abstract. It is known that the coe›cients of the Bernstein expansion of a given multivariate polynomial over a specied box of interest tightly bound the range of the polynomial over the box. e problem of testing the positivity of such a polynomial over a box can thus be reduced to the problem of computing its Bernstein coe›cients. e traditional approach, however, requires that all such coe›cients are computed, and their number is oŸen very large for polynomials with moderately-many variables. A more e›cient method for the implicit representation and computation of Bernstein coe›cients of multivariate polynomials is presented. e complexity becomes nearly linear with respect to the number of terms in the polynomial, instead of exponential with respect to the number of variables. e Bernstein enclosure can also be used to construct a›ne underestimating bound functions for polynomials which can be employed in a branch-and-bound framework for solving constrained global optimization problems. It is known that the coe›cients of the Bernstein expansion of a given multivariate polynomial over a specied box of interest tightly bound the range of the polynomial over the box. e problem of testing the positivity of such a polynomial over a box can thus be reduced to the problem of computing its Bernstein coe›cients. e traditional approach, however, requires that all such coe›cients are computed, and their number is oŸen very large for polynomials with moderately-many variables. A more e›cient method for the implicit representation and computation of Bernstein coe›cients of multivariate polynomials is presented. e complexity becomes nearly linear with respect to the number of terms in the polynomial, instead of exponential with respect to the number of variables. e Bernstein enclosure can also be used to construct a›ne underestimating bound functions for polynomials which can be employed in a branch-and-bound framework for solving constrained global optimization problems. orsten eobald (Frankfurt) Valuations and tropical bases ursday, 11:40 Abstract. Tropical geometry deals with the images of algebraic varieties under real valuations. e resulting tropical varieties are polyhedral cell complexes which preservemany properties of complex algebraic varieties. Among the key concepts in tropical geometry is the one of a tropical basis, which is a basis g , . . . , gm of an ideal I such that the intersection of the tropical hypersurfaces T(g i) coincides with the tropical variety T(I). In this talk, we exhibit the concept of tropical bases, show how to construct short bases by means of regular projections and discuss these projection-based bases. (Based on joint work with Kerstin Hept.) Margaret omas (Oxford) Parameterization and the rational points of denable sets Tuesday, 16:30 Abstract. We provide some background to the problem of bounding the density of rational points We provide some background to the problem of bounding the density of rational points 8 POSITIVITY, VALUATIONS, AND QUADRATIC FORMS lying on transcendental sets, in the context of o-minimal expansions of the reals. In particular, we focus on strategies involving parameterization of denable sets coverings by the images of denable functions with bounded derivatives. We shall look at some recent work towards a conjecture of Wilkie about the real exponential eld and, time permitting, present some results about particular denable sets and whether they do or do not have certain kinds of parameterizations. Jean-Pierre Tignol (Louvain-la-Neuve) Valuations on central simple algebras Saturday, 9:15 Abstract. Valuation theory plays a central role in the solution of various problems concerning nitedimensional division algebras, such as the construction of noncrossed products and of counterexamples to the Kneser-Tits conjecture. However, relating valuations with Brauer-group properties is particularly di›cult because valuations are dened only on division algebras and not on central simple algebras with zero divisors. is talk will present amore žexible tool recently developed in a joint work with AdrianWadsworth, which applies to a broad spectrum of noncommutative situations. In particular, central simple algebras with anisotropic involution over Henselian elds are shown to carry a special kind of value function, which is an analogue of Schilling valuations on division algebras. Valuation theory plays a central role in the solution of various problems concerning nitedimensional division algebras, such as the construction of noncrossed products and of counterexamples to the Kneser-Tits conjecture. However, relating valuations with Brauer-group properties is particularly di›cult because valuations are dened only on division algebras and not on central simple algebras with zero divisors. is talk will present amore žexible tool recently developed in a joint work with AdrianWadsworth, which applies to a broad spectrum of noncommutative situations. In particular, central simple algebras with anisotropic involution over Henselian elds are shown to carry a special kind of value function, which is an analogue of Schilling valuations on division algebras. omas Unger (Dublin) Torsion in Witt groups and sums of hermitian squares Tuesday, 17:15 Abstract. Let F be a formally real eld. It is well-known that the Witt ring W(F) is torsion-free iš every sum of squares in F is again a square in F. For a central simple F-algebra Awith involution σ of the rst kind, consider theWitt groupW(A, σ). One can ask ifW(A, σ) is torsion-free iš every sum of hermitian squares is again a hermitian square. e answer is “yes” in certain simple situations, but “no” in general as I will demonstrate. is is joint work with Vincent Astier. Let F be a formally real eld. It is well-known that the Witt ring W(F) is torsion-free iš every sum of squares in F is again a square in F. For a central simple F-algebra Awith involution σ of the rst kind, consider theWitt groupW(A, σ). One can ask ifW(A, σ) is torsion-free iš every sum of hermitian squares is again a hermitian square. e answer is “yes” in certain simple situations, but “no” in general as I will demonstrate. is is joint work with Vincent Astier. SvenWagner (Konstanz) A decision problem for real multivariate polynomials Friday, 11:30 Abstract. Given any nite sequence h = (h , . . . , hs) of real polynomials in n variables X , . . . , Xn with W(h) = {x ∈ Rn ∣ h(x) ⩾ , . . . , hs(x) ⩾ } non-empty and bounded, we are interested whether every real polynomial f which is strictly positive onW(h) admits a representation f = σ + hσ +⋯ + hsσs where each σ j is a sum of squares of polynomials in R[X , . . . , Xn]. If n = , this always holds, but if n ⩾ , this is not true in general. For n = , Canto Cabral has given an ešective decision procedure that decides whether h has this property. We have shown decidability for every choice of n, and in this talk we want to give a short overview of the proof. Lou van den Dries (Urbana) Immediate extensions of H-elds Saturday, 11:15 Abstract. is is joint work with Aschenbrenner and van der Hoeven. H-elds are ordered dišerential elds where the ordering and derivation interact as in Hardy elds and as in the dišerential eld of logarithmic-exponential series. To obtain a model theory of these valued dišerential elds we need to understand their immediate extensions. is summer wemade progress in this direction, by showing that if K is a Liouville closed H-eld, then the maximal immediate extension of the underlying valued eld of K can be made into an H-eld extension of K. is is joint work with Aschenbrenner and van der Hoeven. H-elds are ordered dišerential elds where the ordering and derivation interact as in Hardy elds and as in the dišerential eld of logarithmic-exponential series. To obtain a model theory of these valued dišerential elds we need to understand their immediate extensions. is summer wemade progress in this direction, by showing that if K is a Liouville closed H-eld, then the maximal immediate extension of the underlying valued eld of K can be made into an H-eld extension of K.