Arithmetic Degree Research Articles

Prime filtrations of monomial ideals and polarizations

We show that an arbitrary monomial ideal I is pretty clean if and only if its polarization I p is clean. This yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree, and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is (non-pure) shellable. We also discuss Stanley decompositions in relation to prime filtrations.

Extended degree functions and monomial modules

The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of Gunston and Vasconcelos, respectively. Particular attention is given to the class of sequentially Cohen-Macaulay modules. The results in this case lead to an algorithm that computes the smallest extended degree.

Arithmetic degree and associated graded modules

We prove that the arithmetic degree of a graded or local ring A is bounded above by the arithmetic degree of any of its associated graded rings with respect to ideal I in A. In particular, if Spec(A) is equidimensional and has an embedded component (i.e., A has an embedded associated prime ideal), then the normal cone of Spec(A) along V(I) has an embedded component too. This extends a result of W. M. Ruppert about embedded components of the tangent cone.

Integrals of automorphic Green's functions associatedto Heegner divisors

In the present paper we find explicit formulas for the degrees of Heegner divisors on arithmetic quotients of the orthogonal group O(2, p) and for the integrals of certain automorphic Green’s functions associated with Heegner divisors. The latter quantities are important in the study of the arithmetic degrees of Heegner divisors in the context of Arakelov geometry. In particular, we obtain a different proof and a generalization of results of Kudla relating these quantities to the Fourier coefficients of certain non-holomorphic Eisenstein series of weight 1 + p/2 for the metaplectic group Mp 2 (Z).

Standard pairs and group relaxations in integer programming

The main result of this paper is a non-Buchberger algorithm for constructing initial ideals and Gröbner bases of toric ideals, based on the connections between toric ideals and integer programming. The tools used are those of standard pair decompositions of standard monomials of a toric initial ideal, localizations of such ideals at their associated primes and group relaxations of integer programs. We give an algorithm for constructing standard pair decompositions, provide degree bounds for certain elements in the reduced Gröbner bases of toric ideals, and derive bounds on the arithmetic degree of initial ideals of monomial curves. We also exhibit new results for the localizations of initial ideals arising from toric ideals of codimension two.

On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals

In the first part of this paper we show that the Castelnuovo-Mumford regularity of a monomial ideal is bounded above by its arithmetic degree. The second part gives upper bounds for the Castelnuovo-Mumford regularity and the arithmetic degree of a monomial ideal in terms of the degrees of its generators. These bounds can be formulated for an arbitrary homogeneous ideal in terms of any Grobner basis.

Towards the arithmetic degree of line bundles� on abelian varieties

In this paper we analyze the integral of the star-product of (n+1) Green currents associated to (n+1) global sections of an ample line bundle equipped with a translation invariant metric over an n-dimensional, polarized abelian variety. The integral is shown to equal the logarithm of the Petersson norm of a certain Siegel modular form, which is explicitly described in terms of the given data. This result can be interpreted as evaluating an archimedian height on a family of polarized abelian varieties. The key ingredient to the proof of the main formula is a dd c -variational formula for the integral under consideration. In the case of dimensions n=1,2,3 explicit examples in terms of classical Riemann theta functions are given.

Robust Proximity Queries: An Illustration of Degree-Driven Algorithm Design

In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact-computation paradigm and formalize the notion of degree of a geometric algorithm as a worst-case quantification of the precision (number of bits) to which arithmetic calculation have to be executed in order to guarantee topological correctness. We also propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach where algorithm design is driven also by the degree, we consider the important classical problem of proximity queries in two and three dimensions and develop a new technique for the efficient and robust execution of such queries based on an implicit representation of Voronoi diagrams. Our new technique offers both low degree and fast query time and for 2D queries is optimal with respect to both cost measures of the paradigm, asymptotic number of operations, and arithmetic degree.

Associated Primes and Arithmetic Degrees

Associated Primes and Arithmetic Degrees

Towards a theory of arithmetic degrees

The aim of this paper is to start a systematic investigation of the arithmetic degree of projective schemes as introduced in [1]. One main theme concerns itself with the behaviour of this arithmetic degree under hypersurface sections, see Theorem 2.1. The classical intersection theory only considers the top-dimensional (or isolated) primary components. However, the notion of arithmetic degree involves the new concept of length-multiplicity of embedded primary ideals as considered in [1], [4], [7], [9], [13]. Therefore it is much harder to control the arithmetic

Gödel incompleteness, explicit expressions for complete arithmetic degrees and applications

AbstractWe summarize in an intuitive vein a few concepts from recursion theory and from the theory of formal systems and then state and comment our recent results on the incompleteness of elementary real analysis and its consequences. Their relation to forcing is also dealt with.

Inequalities Relating Degrees of Adjacent Vertices to the Average Degree

The main theorem gives a class of inequalities concerning finite hypergraphs with a fixed number of vertices per edge. One corollary of the general result: if there are edges, then there is an edge such that the geometric mean of the degrees of its vertices is greater than or equal to the arithmetic mean degree of all the vertices. If the hypergraph is not regular, the inequality is strict. Another corollary of the main result is used to derive an inequality for simple graphs relating the number of edges, the number of vertices, and the largest number of triangles based on a fixed edge in the graph. Finally, some results are derived relating the degrees of non-adjacent vertices of a simple graph to the average degree.

Pseudo-jump operators. II: Transfinite iterations, hierarchies and minimal covers

In this paper we introduce a new hierarchy of sets and operators which we call the REA hierarchy for “recursively enumerable in and above”. The hierarchy is generated by composing (possibly) transfinite sequences of the pseudo-jump operators considered in Jockusch and Shore [1983]. We there studied pseudo-jump operators defined by analogy with the Turing jump as ones taking a set A to A ⊕ for some index e. We would now call these 1-REA operators and will extend them to α-REA operators for recursive ordinals α in analogy with the iterated Turing jump operators (A → A(α) for α < and Kleene's hyperarithmetic hierarchy. The REA sets will then, of course, be the results of applying these operators to the empty set. They will extend and generalize Kleene's H sets but will still be contained in the class of set singletons thus providing us with a new richer subclass of the set singletons which, as we shall see, is related to the work of Harrington [1975] and [1976] on the problems of Friedman [1975] about the arithmetic degrees of such singletons. Their degrees also give a natural class extending the class H of Jockusch and McLaughlin [1969] by closing it off under transfinite iterations as well as the inclusion of [d, d′] for each degree d in the class. The reason for the class being closed under this last operation is that the REA operators include all operators and so give a new hierarchy for them as well as the sets. This hierarchy also turns out to be related to the difference hierarchy of Ershov [1968], [1968a] and [1970]: every α-r.e. set is α-REA but each level of the REA hierarchy after the first extends all the way through the difference hierarchy although never entirely encompassing even the next level of the difference hierarchy.

Finitely generated codings and the degrees r.e. in a degree 𝑑

We introduce finitely generated (partial) lattices which can be used to code an arbitrary set D D . Results of Lerman, Shore and Soare are used to embed these lattices in the degrees r.e. in D D . Thus if the degrees r.e. in and above d {\mathbf {d}} are isomorphic to those r.e. in and above c {\mathbf {c}} , d {\mathbf {d}} and c {\mathbf {c}} are of the same arithmetic degree. Similar applications are given to generic degrees and general homogeneity questions.

Upper bounds on locally countable admissible initial segments of a Turing degree hierarchy

AbstractWhere AR is the set of arithmetic Turing degrees, 0(ω) is the least member of {a(2) ∣ a is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's . This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on results of Jensen and is detailed in [3] and [4]. In § 1 we review the basic definitions from [3] which are needed to state the general results.

Subrecursive program schemata I & II: I. Undecidable equivalence problems; II. Decidable equivalence problems

The study of program schemata and the study of subrecursive programming languages are both concerned with limiting program structure in order to permit a more complete analysis of algorithms while retaining sufficiently rich computing power to allow interesting algorithms. In this paper we combine these approaches by defining classes of subrecursive program schemata and investigating their equivalence problems. Since the languages are all subrecursive, any scheme written in any one of them must halt (as long as we assume the basic functions and predicates are all total). Hence equivalence of schemes is the first question of interest we can ask about these languages. We consider schematic versions of various subrecursive programming languages similar to the Loop language. We distinguish between Pre-Loop and Post-Loop languages on the basis of whether the exit condition in an iteration loop is tested before iteration, an in Algol ( Pre -), or after iteration, as in FORTRAN ( Post -). We slow that at the program level all these languages have the same computing power (the primitive recursive functions) and all have unsolvable equivalence problems (of arithmetic degree II 1 0 ). But at the level of schemes, Pre-Loop has an unsolvable equivalence problem, while at least one formulation of Post-Loop has a solvable equivalence problem. If L is a programming language or scheme language, the we denote by E(L) the equivalence problem in L . The basic languages considered are: Loop (≡Pre-Loop) Loop language for primitive recursive functions Post-Loop Post-Loop language for primitive recursive functions Loop ◊ Loop language with restricted conditionals L [ D ,()] Loop schemata over D with identity L ◊ [ D ,()] Loop schemata with conditionals PL [ D ,()] Post-Loop schemata over D PL ◊ [ D ,()] Post-Loop schemata with conditionals P Program (flowchart) schemata P d Program schemata with DO -statements In contrast to (pure) Loop schemata studied previously by the first author, some of these schemata languages contain the identity function so that a pure data transfer, X ← Y , is possible. Moreover, the equivalence algorithms given here are for the special case of linear schemes (to be defined below) with monadic function variables. Linear schemes are designated by placing L before the name of the more general class, thus LL for linear Loop, LPL for linear Post-Loop, etc. In all schemes considered here the functions are monadic, so no special designation of function rank is provided. It is well known that E(P) is recursively unsolvable and E(P) ∈ II 2 0 . We show that E (Loop), E (Post-Loop), E ( L ◊ ) (both with and without the pure data transfer), and E(L) are recursively unsolvable, while E(LPL) is recursively solvable. The extension of the equivalence algorithm for LPL to polyadic functions appears at present to be a tedious but straightforward modification to the monadic algorithm. We are hopeful that a simpler and more generally applicable technique will emerge for demonstrating solvability or unsolvability of this class of equivalence problems. The algorithm and proofs given here are but a crude first step in delimiting this problem.