Prime Power Case Research Articles

Truncations of ordered abelian groups

We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a +, but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments [0, tau ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group {mathbb {Z}} or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Abstract Let $\varepsilon> 0$ be sufficiently small and let $0 < \eta < 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$ . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$ . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

Tverberg's Theorem at 50: Extensions and Counterexamples

We describe how a powerful new “constraint method” yields many different extensions of the topological version of Tverberg’s 1966 Theorem in the prime power case— and how the same method also was instrumental in the recent spectacular construction of counterexamples for the general case. © 2016. All rights reserved.

On Dedekind sums with equal values

Dedekind sums [Formula: see text] occur in many fields of mathematics. Since [Formula: see text] if [Formula: see text], it is natural to ask which of the Dedekind sums [Formula: see text], [Formula: see text], take equal values. So far no simple criterion is known by which the equality of [Formula: see text] and [Formula: see text] could be decided. In this note, we show how to obtain non-obvious examples of equal Dedekind sums. We consider two cases which mark the extreme possibilities for the argument [Formula: see text], namely, [Formula: see text] a prime power and [Formula: see text] square-free. Whereas we can give a partial overview of equal Dedekind sums in the prime power case, such an overview seems to be much more difficult to obtain in the square-free case.

Convex equipartitions via Equivariant Obstruction Theory

We describe a regular cell complex model for the configuration space F(ℝ d , n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.

An abundancy result for the two prime power case and results for an equations of Goormaghtigh

An abundancy result for the two prime power case and results for an equations of Goormaghtigh

Mutually orthogonal Latin squares and mutually unbiased bases in dimensions of odd prime power

There has been much interest in mutually unbiased bases (MUBs) and their connections with various other discrete structures, such as projective planes, mutually orthogonal Latin squares (MOLS) etc. It has been conjectured by Saniga et al. (J Opt B Quantum Semiclass Opt 6:L19---L20, 2004) that the existence of a complete set of MUBs in ? d is linked to the existence of a complete set of MOLS of side length d. Since more is known about MOLS than about MUBs, most research has concentrated on constructing MUBs from MOLS (Roy and Scott, J Math Phys 48:072110, 2007; Paterek et al., Phys Rev A 70:012109, 2009). This paper gives a simple algebraic construction of MOLS from two known constructions of MUBs in the odd prime power case.

Block-circulant matrices with circulant blocks, Weil sums, and mutually unbiased bases. II. The prime power case

In our previous paper [Combescure, M., “Circulant matrices, Gauss sums and the mutually unbiased bases. I. The prime number case,” Cubo A Mathematical Journal (unpublished)] we have shown that the theory of circulant matrices allows to recover the result that there exists p+1 mutually unbiased bases in dimension p, p being an arbitrary prime number. Two orthonormal bases B, B′ of Cd are said mutually unbiased if ∀b∊B, ∀b′∊B′ one has that |b⋅b′|=1/d (b⋅b′ Hermitian scalar product in Cd). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if d=pn (p a prime number and n any integer) there exists d+1 mutually unbiased bases in Cd. Our result relies heavily on an idea of Klimov et al. [“Geometrical approach to the discrete Wigner function,” J. Phys. A 39, 14471 (2006)]. As a subproduct we recover properties of quadratic Weil sums for p≥3, which generalizes the fact that in the prime case the quadratic Gauss sum properties follow from our results.

Equiangular tight frames from Paley tournaments

We prove the existence of equiangular tight frames having n = 2 d - 1 elements drawn from either C d or C d - 1 whenever n is either 2 k - 1 for k ∈ N , or a power of a prime such that n ≡ 3 mod 4 . We also find a simple explicit expression for the prime power case by establishing a connection to a 2 d -element equiangular tight frame based on quadratic residues.

On the number of Tverberg partitions in the prime power case

We give an extension of the lower bound of A. Vućić, R. Živaljević [Notes on a conjecture of Sierksma, Discrete Comput. Geom. 9 (1993) 339–349] for the number of Tverberg partitions from the prime to the prime power case. Our proof is inspired by the Z p -index version of the proof in [J. Matoušek, Using the Borsuk–Ulam Theorem, in: Lectures on Topological Methods in Combinatorics and Geometry, Universitext, Springer-Verlag, Heidelberg, 2003] and uses Volovikov’s Lemma. Analogously, one obtains an extension of the lower bound for the number of different splittings of a generic necklace to the prime power case.

On isomorphisms of abelian Cayley objects of certain orders

Let m be a positive integer such that gcd( m, ϕ( m))=1 ( ϕ is Euler's phi function) with m= p 1⋯ p r the prime power decomposition of m. Let n= p 1 a 1 ⋯ p r a r . We provide a sufficient condition to reduce the Cayley isomorphism problem for Cayley objects of an abelian group of order n to the prime power case. In the case of Cayley k-ary relational structures (which include digraphs) of abelian groups, this sufficient condition reduces the Cayley isomorphism problem of k-ary relational structures of abelian groups to the prime power case for Cayley k-ary relational structures of abelian groups. As corollaries, we solve the Cayley isomorphism problem for Cayley graphs of Z n (for the specific values of n as above) and prove several abelian groups (for specific choices of the a i ) of order n are CI-groups with respect to digraphs.

Class Number One Problem for Imaginary Function Fields: The Cyclic Prime Power Case

In this paper, we determine all finite separable imaginary extensions K/Fq(x) whose maximal order is a principal ideal domain in case K/Fq(x) is a non zero genus cyclic extension of prime power degree. There exist exactly 42 such extensions, among which 7 are non isomorphic over Fq.