波兰科学研究院数学研究所
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作者:Alex Rice
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.187, pp.1-41IMPAS
摘要:We show that if $h\in \mathbb Z[x]$ is a polynomial of degree $k\geq 2$ such that $h(\mathbb N)$ contains a multiple of $q$ for every $q\in\mathbb N$, known as an intersective polynomial , then any subset of $\{1,\dots,N\}$ with no nonzero differences of the form $h(n)$ for ...
作者:Jeremy Rouse
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.187, pp.81-100IMPAS
摘要:Let $Q$ be a positive-definite quaternary quadratic form with integer coefficients. We study the problem of giving bounds on the largest positive integer $n$ that is locally represented by $Q$ but not represented. Assuming that $n$ is relatively prime to $D(Q)$, the determinant o...
作者:Steve Balady , Lawrence C. Washington
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.187, pp.43-57IMPAS
摘要:We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.
作者:Heng Huat Chan , Liuquan Wang
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.187, pp.59-80IMPAS
摘要:Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$.It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfiescongruences such as $p(5n+4)\equiv 0\pmod{5}$. In this article, we discusscongruences sat...
作者:Grzegorz Banaszak , Cristian Popescu
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.188, pp.125-169IMPAS
摘要:Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer–Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$-equivariant Hecke characters for $F$ with infinite type equal to a spec...
作者:Carlos Matheus , Carlos Gustavo Moreira
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.188, pp.183-208IMPAS
摘要:The complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ was studied by many authors (including Freiman, Berstein, Cusick and Flahive). From their works, we know a countable collection of points in $M\setminus L$.;;In this article, we describe the str...
作者:Pierfrancesco Carlucci
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.187, pp.301-327IMPAS
摘要:We find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{\rm ns}(p^k) $ associated to the normalizer of a non-split Cartan subgroup of level $ p^k $ where $ p\not=2,3 $ is a prime. The cuspidal divisor class group $ \mathfrak{C}^+...
作者:C. Miguel
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.188, pp.317-324IMPAS
摘要:For a finite commutative ring $R$ with identity, we obtain an exact formula for the number of ways to represent each element of $R$ as the sum of two exceptional units. This generalizes to finite rings a recent result of J. W. Sander for the ring $\mathbb Z_n$ of residue classes ...
作者:László Mérai , Igor E. Shparlinski
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.188, pp.401-411IMPAS
摘要:For a prime $p$ and a polynomial $F(X,Y)$ over the finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions of$$F(x,y)=0, \quad x\in\mathcal{A}, \, y\in \mathcal{B},$$where $\mathcal A$ and $\mathcal B$ are very small intervals or subgroups. Th...
作者:Ralph Kritzinger
来源:[J].Acta Arithmetica(IF 0.472), 2019, Vol.187, pp.151-187IMPAS
摘要:We use the Haar function system in order to study the $L_2$ discrepancy of a class of digital $(0,n,2)$-nets. Our approach yields exact formulas for this quantity, which measures the irregularities of distribution of a set of points in the unit interval. We obtain such formu...

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