Related papers: Counting multi-quadratic number fields of bounded …

In this paper we show that there exist infinitely many square-free numbers of the form $n^2+n+1$. We achieve this by deriving an asymptotic formula by improving the reminder term from previous results.

We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…

For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and…

This paper presents fundamental algorithms for the computational theory of quadratic forms over number fields. In the first part of the paper, we present algorithms for checking if a given non-degenerate quadratic form over a fixed number…

We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.

We implement and interpret various supervised learning experiments involving real quadratic fields with class numbers 1, 2 and 3. We quantify the relative difficulties in separating class numbers of matching/different parity from a…

We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.

We show how the Selberg $\Lambda^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of…

Two types of errors can occur when discriminating pairs of quantum states. Asymmetric state discrimination involves minimizing the probability of one type of error, subject to a constraint on the other. We give explicit expressions bounding…

We enumerate all totally real number fields F with root discriminant delta_F <= 14. There are 1229 such fields, each with degree [F:QQ] <= 9.

We calculate the asymptotic value of the choice number of complete multi-partite graphs.

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…

In this paper we are interested in the class numbers of a family of real quadratic fields for which the square roots of the discriminants have a known expansion in continued fraction. In particular we prove that $h(D)>1$, with possibly a…

Using the circle method, we show that for a fixed positive definite integral quadratic form $A$, the expected asymptotic formula for the number of representations of a positive definite integral quadratic form $B$ by $A$ holds true,…

Using the circle method, we obtain asymptotic formulae for the number of integer solutions to certain quadratic polynomials that are uniform in the coefficients of the polynomial.

Asymptotic expressions for Clebsch-Gordan coefficients are derived from an exact integral representation. Both the classically allowed and forbidden regions are analyzed. Higher-order approximations are calculated. These give, for example,…

It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders…

Asymptotic formulae are established for the number of natural numbers $m$ with largest square-free divisor not exceeding $m^{\vartheta}$, for any fixed positive parameter $\vartheta$. Related counting functions are also considered.

We solve unconditionally the class number one problem for the $2$-parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ with square-free discriminant $d=(an)^2+4a$ for positive odd integers $a$ and $n$.

In this paper, we construct certain infinite families of imaginary quadratic fields whose class number is divisible by a given positive integer.