When is scalar multiplication decidable?

Let K be a subfield of $\mathbb{R}$. The theory of $\mathbb{R}$ viewed as an ordered K-vector space and expanded by a predicate for $\mathbb{Z}$ is decidable if and only if K is a real quadratic field.     

Partial augmentations power property: A Zassenhaus Conjecture related problem

Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group G is conjugate in the rational group algebra of G to an element in ±G. We review the known weaker versions of this conjecture and introduce a new condition, on the partial augmentations of the powers of a unit of finite order in $\mathbb{Z}G$, which is weaker than the Zassenhaus Conjecture but stronger than its other weaker versions.We prove that this condition is satisfied for units mapping to the identity modulo a nilpotent normal subgroup of G. Moreover, we show that if the condition holds then the HeLP Method adopts a more friendly form and use this to prove the Zassenhaus Conjecture for a special class of groups.     

The extension theorem for the Lee and Euclidean weights over Z/pkZ

The Lee and Euclidean weights have the extension property over the local rings $\mathbb{Z}/p^k\mathbb{Z}$, p prime. The non-vanishing of certain Fourier coefficients is established by expressing the coefficients in terms of generalized Bernoulli numbers and making use of knowledge of the locations of zeros of Dirichlet L-functions.     

On Suslin homology with integral coefficients in characteristic zero (with an appendix by Bruno Kahn)

We show that the Suslin homology group with integral coefficients of a scheme X separated of finite type over an algebraically closed field of characteristic 0 is a direct sum of a uniquely divisible group, finite copies of $\mathbb{Q}/\mathbb{Z}$, and a finitely generated group. We also study the possible type of homomorphisms between such groups induced by the morphisms of schemes. An appendix written by Bruno Kahn is included, which simplifies the proofs and generalizes the results.     

Construction of extremal Type II Z2k-codes

We give methods for constructing many self-dual ${\mathbb{Z}}_m$-codes and Type II ${\mathbb{Z}}_{2k}$-codes of length 2n starting from a given self-dual ${\mathbb{Z}}_m$-code and Type II ${\mathbb{Z}}_{2k}$-code of length 2n, respectively. As an application, we construct extremal Type II ${\mathbb{Z}}_{2k}$-codes of length 24 for $k=4,5,\dots ,20$ and extremal Type II ${\mathbb{Z}}_{2k}$-codes of length 32 for $k=4,5,\dots ,10$. We also construct new extremal Type II ${\mathbb{Z}}_4$-codes of lengths 56 and 64.     

Integral representations of martingales for progressive enlargements of filtrations

We work in the setting of the progressive enlargement $\mathbb{G}$ of a reference filtration $\mathbb{F}$ through the observation of a random time $\tau$. We study an integral representation property for some classes of $\mathbb{G}$-martingales stopped at $\tau$. In the first part, we focus on the case where $\mathbb{F}$ is a Poisson filtration and we establish a predictable representation property with respect to three $\mathbb{G}$-martingales. In the second part, we relax the assumption that $\mathbb{F}$ is a Poisson filtration and we assume that $\tau$ is an $\mathbb{F}$-pseudo-stopping time. We establish integral representations with respect to some $\mathbb{G}$-martingales built from $\mathbb{F}$-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two $\mathbb{G}$-martingales.     

Characterizing immutable sandpiles: A first look

By working with coefficients in $\mathbb{Z}$ or $\mathbb{R}$, one can define two different notions of stability for a sandpile on a graph. We call a sandpile immutable when these notions agree. Our main results give linear-algebraic characterizations for large classes of immutable sandpiles.     

Construction of the circle in UniMath

We show that the type $\mathrm{T}\mathbb{Z}$ of $\mathbb{Z}$-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence Axiom and propositional truncation, yielding a stand-alone construction of the circle not using higher inductive types.     

The cotype zeta function of Zd

We give an asymptotic formula for the number of sublattices $\Lambda \subseteq {\mathbb{Z}}^d$ of index at most $X$ for which ${\mathbb{Z}}^d/\Lambda$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to work of Stanley and Wang on Smith normal forms of random integral matrices and discuss connections to the Cohen–Lenstra heuristics. Our arguments are based on Petrogradsky’s formulas for the cotype zeta function of ${\mathbb{Z}}^d$, a multivariable generalization of the subgroup growth zeta function of ${\mathbb{Z}}^d$.     

On the class semigroup of ZS-fields and Iwasawa invariants

Let p be a prime number. In [15], we studied the class semigroup of the ring of integers of the cyclotomic ${\mathbb{Z}}_p$-extension of the rationals. In this paper, we generalize the result to some ${\mathbb{Z}}_S$-extensions of number fields. Moreover, we investigate the relation between the class semigroup and Iwasawa invariants.     

Fq-linear codes over Fq-algebras

Huffman (2013) [12] studied ${\mathbb{F}}_q$-linear codes over ${\mathbb{F}}_{q^m}$ and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative ${\mathbb{F}}_q$-algebra. An ${\mathbb{F}}_q$-linear code over S of length n is an ${\mathbb{F}}_q$-submodule of $S^n$. In this paper, we study ${\mathbb{F}}_q$-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over ${\mathbb{F}}_q$-algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of ${\mathbb{F}}_q$-linear codes over finite commutative graded ${\mathbb{F}}_q$-algebras.     

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup $C{l}_1(\mathbb{Z}P)$ of $S{K}_1(\mathbb{Z}P)$, in terms of a genetic basis of P. We also introduce a deflation map $C{l}_1(\mathbb{Z}P)\to C{l}_1(\mathbb{Z}(P/N))$, for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $S{K}_1(\mathbb{Z}P)$, when P is an elementary abelian p-group.