Field Of Puiseux Series Research Articles

Degenerated Calabi–Yau varieties with infinite components, moduli compactifications, and limit toroidal structures

For any degenerating Calabi–Yau family, we introduce a new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi–Yau varieties, parametrized by the rational points of the Kontsevich–Soibelman’s essential skeleton, while dominated by the Huber adification over the Puiseux series field. Other topics include: projective limits of toroidal compactifications (Sect. 3), locally modelled on limit toric varieties (Sect. 2.4), the way to attach a tropicalized family to a given Calabi–Yau family (Sect. 4), which are weakly related to each other.

Necessary and sufficient conditions for the existence of invariant algebraic curves

We present a set of conditions enabling a polynomial system of ordinary differential equations in the plane to have invariant algebraic curves. These conditions are necessary and sufficient. Our main tools include factorizations over the field of Puiseux series near infinity of bivariate polynomials generating invariant algebraic curves. The set of conditions can be algorithmically verified. This fact gives rise to a method, which is able not only to find some irreducible invariant algebraic curves, but also to perform their classification. We study in details the problem of classifying invariant algebraic curves in the most difficult case: we consider differential systems with infinite number of trajectories passing through infinity. As an example, we find necessary and sufficient conditions such that a general polynomial Lienard differential system has invariant algebraic curves. We present a set of all irreducible invariant algebraic curves for quintic Lienard differential systems with a linear damping function. It is supposed in scientific literature that the degrees of their irreducible invariant algebraic curves are bounded by 6. While we derive irreducible invariant algebraic curves of degree 9.

The tropical analogue of the Helton–Nie conjecture is true

Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers can be written as the projection of a spectrahedron. Recently, Scheiderer disproved this conjecture. We show, however, that the following result, which may be thought of as a tropical analogue of this conjecture, is true: over a real closed nonarchimedean field of Puiseux series, the convex semialgebraic sets and the projections of spectrahedra have precisely the same images by the nonarchimedean valuation. The proof relies on game theory methods.

Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group

Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, o-minimal geometry and valuation theory. We set up the construction in such a way that it is determined by a section of the valuation. If the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism. The range of the measure and integration is obtained in a controled and tame way from the real closed field we start with. The main example is given by the case of the field of Puiseux series where the range is the polynomial ring in one variable over this field.

Tropicalization of facets of polytopes

It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu.

Incomplete LU Preconditioner Based on Max-Plus Approximation of LU Factorization

We present a new method for the a priori approximation of the orders of magnitude of the entries in the LU factors of a complex or real matrix $A$. This approximation is used to determine the positions of the largest entries in the LU factors of $A$, and these positions are used as the sparsity pattern for an incomplete LU factorization preconditioner. Our method uses max-plus algebra and is based solely on the moduli of the entries of $A$. We also present techniques for predicting which permutation matrices will be chosen by Gaussian elimination with partial pivoting. We exploit the strong connection between the field of Puiseux series and the max-plus semiring to prove properties of the max-plus LU factors. Experiments with a set of test matrices from the University of Florida Sparse Matrix Collection show that our max-plus LU preconditioners outperform traditional level of fill methods and have similar performance to those preconditioners computed with more expensive threshold-based methods.

Non-archimedean valuations of eigenvalues of matrix polynomials

We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities become equalities under genericity conditions, and that the leading coefficients of the eigenvalues are determined as the eigenvalues of auxiliary matrix polynomials.

P-adically closed fields with nonstandard analytic structure

AbstractWe prove quantifier elimination for the field ℚp((tℚ)) (the completion of the field of Puiseux series over ℚp) in Macintyre's language together with symbols for functions in a class containing both t-adically and p-adically overconvergent functions. We also show that the theory of ℚp((tℚ)) is b-minimal in this language.

The j-invariant of a plane tropical cubic

In this paper, we prove that for a plane cubic over the field of Puiseux series, the negative of the generic valuation of the j-invariant is equal to the cycle length of the tropicalization of the curve, if there is a cycle at all.

Real closed fields with non-standard and standard analytic structure

We consider the ordered field which is the completion of the Puiseux series field over ℝ equipped with a ring of analytic functions on [−1, 1]n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures of Denef and van den Dries [Ann. of Math. 128 (1988) 79–138] and Lipshitz and Robinson [Bull. London Math. Soc. 38 (2006) 897–906]. We prove quantifier elimination and o-minimality in the corresponding language. We extend these constructions and results to rank n ordered fields ℝn (the maximal completions of iterated Puiseux series fields). We generalize the example of Hrushovski and Peterzil [J. Symbolic Logic 72 (2007) 119–122] of a sentence which is not true in any o-minimal expansion of ℝ (shown in [Bull. London Math. Soc. 38 (2006) 897–906] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences σn, true in ℝn, but not true in any o-minimal expansion of any of the fields ℝ, ℝ1, …, ℝn−1.

Standard bases in [formula omitted

In this paper we study standard bases for submodules of K [ [ t 1 , … , t m ] ] [ x 1 , … , x n ] s respectively of their localisation with respect to a t ¯ -local monomial ordering. The main step is to prove the existence of a division with remainder generalising and combining the division theorems of Grauert–Hironaka and Mora. Everything else then translates naturally. Setting either m = 0 or n = 0 we get standard bases for polynomial rings respectively for power series rings as a special case. We then apply this technique to show that the t -initial ideal of an ideal over the Puiseux series field can be read of from a standard basis of its generators. This is an important step in the constructive proof that each point in the tropical variety of such an ideal admits a lifting.

OVERCONVERGENT REAL CLOSED QUANTIFIER ELIMINATION

Let K be the (real closed) field of Puiseux series in t over R endowed with the natural linear order. Then the elements of the formal power series rings R[[ξ1, . . . , ξn]] converge t-adically on [−t, t]n, and hence define functions [−t, t]n → K. Let L be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.17, K is o-minimal in L. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

Un analogue non commutatif du theoreme de puiseux

Let K be a field, and δ a derivation of K. The right zeros of skew polynomials in the formal differential operator ring K[T,δ], with commutation law Tα - αT = δ(α) for any α ∊ K, are logarithmic derivatives of solutions of linear differential equations in some differential extension of K. For K the Puiseux series field over an algebraically closed field of characteristic zero, and δ the classical derivation, we prove that any polynomial of degree > 1 in K[T,δ] splits, without unicity, into factors of degree one.