Nonstandard quadratic forms over rational function fields

Tensors associated with a pair of quadratic differential forms

Canonical dimension of a smooth complete connected variety is the minimal dimension of image of its rational endomorphism. The $i$-th canonical dimension of a non-degenerate quadratic form is the canonical dimension of its $i$-th orthogonal grassmannian. The maximum of a canonical dimension for quadratic forms of a fixed dimension is known to be equal to the dimension of the corresponding grassmannian. This article is about the minima of the canonical dimensions of an anisotropic quadratic form. We conjecture that they equal the canonical dimensions of an excellent anisotropic quadratic form of the same dimension and we prove it in a wide range of cases.

In this chapter, we discuss the general theory of quadratic forms and the notion of congruence and its invariants, as well as the applications of the theory to the analysis of special forms. We will focus primarily on quadratic forms induced by rational functions, most notably the Hankel and Bezout forms, because of their connection to system-theoretic problems such as stability and signature symmetric realizations. These forms use as their data different representations of rational functions, namely power series and coprime factorizations respectively. But we will discuss also the partial fraction representation in relation to the computation of the Cauchy index of a rational function and the proof of the Hermite–Hurwitz theorem and the continued fraction representation as a tool in the computation of signatures of Hankel matrices as well as in the problem of Hankel matrix inversion. Thus, different representations of rational functions, i.e., different encodings of the information carried by a rational function, provide efficient starting points for different methods. The results obtained for rational functions will be applied to root-location problems for polynomials in the next chapter.

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

We present a well-parallelizable algorithm which, taking a straight-line program for the evaluation of a vectorial field of rational functions of Q(X 1,...,X n ) as input, decides whether they allow a rational potential function and, in case of affirmative answer, computes it as output. We introduce a mixed model of representation of polynomials to allow the application of integration techniques and show how to perform some basic operations with it. The algorithm is presented as a family of arithmetic networks of polynomial size and poly logarithmic depth in the degree of the occurring polynomials.KeywordsRational FunctionPartial FractionGreat Common DivisorProbabilistic AlgorithmParallel TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Isotropy of Six-Dimensional Quadratic Forms over Function Fields of Quadrics

This dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.

Biquaternion division algebras and fourth power-central elements

In this paper we consider the problem of sparse signal modeling by means of rational functions. Our dictionary is composed by a finite collection of elementary rational functions. In order to represent the signal with minimal error, we select an optimal number of basis from this set. The mutual coherence is a fundamental attribute of the dictionary. We analyze this quantity and describe its relation to the free parameters, i.e., the inverse poles, of rational functions. Then, we demonstrate the efficiency of sparse rational representations by compressing real electrocardiograms (ECG) including comparisons with other methods.

According to the theory of electrical networks, we can explicitly get the state equation for a time-invariant RLC network. A computer analysis program is compiled for finding the state equation of RLC network over the field of the rational function in multi-parameters. The program has no limits to the orders of state equation and the complex degree of RLC network. It is operated easily and obtains a good interface.

We study the ``imaginary" binary quadratic form equations ax^2+bxy+cy^2+g=0 over k[t] in rational function fields, showing that a condition with respect to the Artin reciprocity map, is the only obstruction to the local-global principle for integral solutions of the equation.

Sur les dimensions des formes quadratiques de hauteur 2

Hilbertʼs Tenth Problem for rational function fields over p-adic fields

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, k denotes a fixed algebraically closed field. Sections 1–6 establish the definitions and elementary properties of varieties, maps between varieties, and dimension, all over k. Sections 1–3 concern varieties and dimension. Affine algebraic sets, affine varieties, and the Zariski topology on affine space are introduced in Section 1, and projective algebraic sets and projective varieties are introduced in Section 3. Section 2 defines the geometric dimension of an affine algebraic set, relating the notion to Krull dimension and transcendence degree. The actual context of Section 2 is a Noetherian topological space, the Zariski topology on affine space being an example. In such a space every closed subset is the finite union of irreducible closed subsets, and the union can be written in a certainway thatmakes the decomposition unique. Every nonempty closed set has a meaningful geometric dimension. In affine space the irreducible closed sets are the varieties, and each variety acquires a geometric dimension. The discussion in Section 2 applies in the context of projective space as well, and thus each projective variety acquires a geometric dimension. Moreover, any nonempty open subset of a Noetherian space is Noetherian. A nonempty open subset of an affine variety is called quasi-affine, and a nonempty open subset of a projective variety is called quasiprojective. Each quasi-affine variety or quasiprojective variety has a dimension equal to that of its closure, which is a variety. Sections 4–6 take up maps between varieties. Section 4 introduces spaces of scalar-valued functions on quasiprojective varieties–rational functions, functions regular at a point, and functions regular on an open set. The section goes on to relate these notions for the different kinds of varieties. Section 5 introduces morphisms, which are a restricted kind of function between varieties. The tools of Sections 4–5 together show that for many purposes all the different kinds of varieties can be treated as quasiprojective varieties. Section 6 introduces rational maps between varieties; these are not everywhere-defined functions, but each can be restricted to an open dense subset on which it is a morphism. Rational maps with dense image correspond to field mappings of the fields of rational functions, with the order of the mappings reversed. Section 7 concerns singularities at points of varieties, still over the field k. Zariski's Theorem was stated in Chapter VII for affine varieties and partly proved at that time. In the current context it has a meaning for any point of any quasiprojective variety. The section proves the full theorem, which characterizes singular points in a way that shows they remain singular under isomorphisms of varieties. Section 8 concerns classification questions over k for irreducible curves, i.e., quasiprojective varieties of dimension 1. From Section 6 it is known that two irreducible curves are equivalent under rational maps if and only if their fields of rational functions are isomorphic. The main theorem of Section 8 is that each such equivalence class of irreducible curves contains an everywhere nonsingular projective curve, and this curve is unique up to isomorphism of varieties. The points of this curve are parametrized by those discrete valuations of the underlying function field that are defined over k. Sections 9–12 relate the general theory of Sections 1–6 to the topic of solutions of simultaneous solutions of polynomial equations, as treated at length in Chapter VIII. Section 9 treats monomial ideals in k[X1, . . . , Xn], identifying their zero loci concretely and computing their dimension. The section goes on to introduce the affineHilbert function of this ideal, which measures the proportion of polynomials of degree≤ s not in the ideal. In the way that this function is defined, it is a polynomial for large s called the affine Hilbert polynomial of the ideal. Its degree equals the dimension of the zero locus of the ideal. Section 10 extends this theory from monomial ideals to all ideals, again concretely computing the dimension of the zero loci, obtaining an affine Hilbert polynomial, and showing that its degree equals the dimension of the zero locus of the ideal. Section 11 adapts the theory to homogeneous ideals and projective algebraic sets by making use of the cone in affine space over the set in projective space. Section 12 applies the theory of Section 11 to address the question how the dimension of a projective algebraic set is cut down when the set is intersected with a projective hypersurface. A consequence of the theory is the result that a homogeneous system of polynomial equations over an algebraically closed field with more unknowns than equations has a nonzero solution. Section 13 is a brief introduction to the theory of schemes, which extends the theory of varieties by replacing the underlying algebraically closed field by an arbitrary commutative ring with identity.