Function fields and places

The third chapter contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface. For such surfaces, the geometry of ramified coverings and Galois theory are not only analogous but in fact very closely related to each other. This relationship is useful in both directions. On the one hand, Galois theory and Riemann’s existence theorem allow one to describe the field of functions on a ramified covering over a Riemann surface as a finite algebraic extension of the field of meromorphic functions on the Riemann surface. On the other hand, the geometry of ramified coverings together with Riemann’s existence theorem allows one to give a transparent description of algebraic extensions of the field of meromorphic functions over a Riemann surface.

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.

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.

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.

Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class NC. The factorization results also extend to the case where the coefficient field is a function field.

In the classical theory of multiple zeta values (MZV’s), Furusho proposed a conjecture asserting that the p-adic MZV’s satisfy the same \({\mathbb {Q}}\)-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho’s conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations. To each tuple of positive integers \({\mathfrak {s}}=(s_1, \ldots , s_r)\), we construct a corresponding t-module together with a specific rational point. The fine resolution (via fiber coproduct) of this construction actually allows us to obtain nice logarithmic interpretations for both the \(\infty \)-adic MZV and v-adic MZV at \({\mathfrak {s}}\), completely generalizing the work of Anderson–Thakur (Ann Math (2) 132(1):159–191, 1990) in the case of \(r=1\). Furthermore it enables us to apply Yu’s sub-t-module theorem (Yu in Ann Math (2) 145(2):215–233, 1997), connecting any \(\infty \)-adic linear relation on MZV’s with a sub-t-module of a corresponding giant t-module. This makes it possible to arrive at the same linear relation for v-adic MZV’s.

A characterization of a linear lumped constant network analysis is made by the concept of semibasis. The semibasis <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> , over a coefficient set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> (of rational functions of edge immittances over the real field) is a set of variables such that any variable of the network can be represented by or whole the analysis equations can be written by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> , over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> . Graph theoretical conditions of the semibasis over certain familiar coefficient sets such as the field of rational functions, the ring of polynomials, and the sets of quadratic or linear polynomials are given. Considerations on the minimum semibasis, the cardinality of which is called the graph theoretical degree of freedom, over the linear or quadratic polynomial coefficient sets are made. Some graph theoretical unsolved problems related to the principal partition of a graph and the central tree are shown to be essential.

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a “free” polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.

The fields of rational functions on the moduli varieties of stable (symplectic) vector bundles over the projective plane are described. As a consequence the rationality of some series of moduli varieties is proved.

1. Statement of results We let F be the finite field with two elements, and let K be the field F((x')) of all formal power series in x-' over F. Every element of K not in the field of rational functions F(x) has a unique infinite continued fraction expansion f = a. + 1/(al + 1/(a2 + * * *)), or more briefly f = [a0; a,, a2, * *.* ],

We study singularly perturbed linear programs. These are parametric linear programs whose constraints become linearly dependent when the perturbation parameter goes to zero. Problems like that were studied by Jeroslow (1973). He proposed simplex-like method, which works over the field of rational functions. Here we develop an alternative asymptotic simplex method based on Laurent series expansions. This approach appears to be more computationally efficient. In addition, we point out several possible generalizations of our method and provide new simple updating formulae for the perturbed solution.

We consider the problem of simultaneous approximation to a number and to its square in a general framework that encompasses imaginary quadratic number fields and fields of rational functions in one variable. In this context, we construct new extremal numbers.

An effective method for computing parametric standard bases of Jacobian ideals is introduced for semi-weighted homogeneous isolated hypersurface singularities. The advantage is that the proposed method is algorithmically simple. The main ideas of the method are the use of a negative weighted term ordering and coefficients in a field of rational functions. The correctness of the method is proved by utilizing algebraic local cohomology classes associated to semi-weighted homogeneous singularities.

A rational function whose denominator polynomial has no zeros inside the unit circle, is called a generalized polynomial, and has properties similar to a polynomial. The ring of generalized polynomials is denoted by R∑(σ). A generalized polynomial matrix has the ∑-Smith form similar to the Smith form of a polynomial matrix. Linear neutral time-delay systems can be represented as systems over the field of rational functions whose argument is σ, where σ is a fixed delay operator. R∑(σ)-controllability which is similar to R[σ]-controllability for a time-delay system of the retarded type, is defined. Theorem 1 gives a necessary and sufficient condition for R∑(σ)-controllability. This condition is represented by the ∑-Smith form for the controllability matrix.∑-pole assignability which is the extension of pole assignability for a retarded system and guarantees asymptotic stability of the closed loop system by phiscally realizable state feedback, is defined. Theorem 2 states that R∑(σ)-controllability is equivalent to ∑-pole assignability. Theorem 3 gives a sufficient condition for decoupling by physically realizable state feedback. Finally, it is shown that under the condition of Theorem 3 and R∑(σ)-controllability, there exists a differential-difference compensator of the neutral type to achieve decoupling and arbitrary ∑-pole assignment.

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