In Computer Algebra, Subresultant Theory provides a powerful method to construct algorithms solving problems for polynomials in one variable in an optimal way. So, using this method we can compute the greatest common divisor of two polynomials in one variable with integer coefficients avoiding the exponential growth of the coefficients that will appear if we use the Euclidean Algorithm.In this note, generalizing a forgotten construction appearing in [Hab], we extend the Subresultant Theory to the...

We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

We show how to adapt Terr’s variant of the baby-step giant-step algorithm of Shanks to the computation of the regulator and of generators of principal ideals in real-quadratic number fields. The worst case complexity of the resulting algorithm depends only on the square root of the regulator, and is smaller than that of all other previously specified unconditional deterministic algorithm for this task.

We prove an explicit bound for N(σ,T), the number of zeros of the Riemann zeta function satisfying ℜ𝔢 s ≥ σ and 0 ≤ ℑ𝔪 s ≤ T. This result provides a significant improvement to Rosser's bound for N(T) when used for estimating prime counting functions.

This paper presents algorithms for quadratic forms over a formally real algebraic function field K of one variable over a fixed real closed field k. The algorithms introduced in the paper solve the following problems: test whether an element is a square, respectively a local square, compute Witt index of a quadratic form and test if a form is isotropic/hyperbolic. Finally, we remark on a method for testing whether two function fields are Witt equivalent.

The famous problem of determining all perfect powers in the Fibonacci sequence ${\left(F_n\right)}_{n\ge 0}$ and in the Lucas sequence ${\left(L_n\right)}_{n\ge 0}$ has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations $L_n=q^a{y}^p$, with $a\>0$ and $p\ge 2$, for all primes $q\<1087$ and indeed for all but $13$ primes $q\<{10}^6$. Here the strategy of [10] is not sufficient due to the sizes of...

Zeta-generalized-Euler-constant functions, $$\gamma \left(s\right):=\sum _{k=1}^{\infty }\left(\frac{1}{k^s}-{\int }_k^{k+1}\frac{dx}{x^s}\right)$$ and $$\tilde{\gamma }\left(s\right):=\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}\left(\frac{1}{k^s}-{\int }_k^{k+1}\frac{dx}{x^s}\right)$$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $$\tilde{\gamma }$$ (1) = ln $$\frac{4}{\pi }$$ , are studied and estimated with high accuracy.

An asymptotic approximation of Wallis’ sequence W(n) = Πk=1n 4k 2/(4k 2 − 1) obtained on the base of Stirling’s factorial formula is presented. As a consequence, several accurate new estimates of Wallis’ ratios w(n) = Πk=1n(2k−1)/(2k) are given. Also, an asymptotic approximation of π in terms of Wallis’ sequence W(n) is obtained, together with several double inequalities such as, for example, $W\left(n\right)·(a_n+b_n)<\pi <W\left(n\right)·(a_n+b_n^{\text{'}})$ with $a_n=2+\frac{1}{2n+1}+\frac{2}{3{(2n+1)}^2}-\frac{1}{3n{(2n+1)}^{\text{'}}}{b}_n=\frac{2}{33{(n+1)}^{2^{\text{'}}}}{b}_n^{\text{'}}\frac{1}{13n^{2^{\text{'}}}}n\in \mathbb{N}$ .

We give an effective procedure to find minimal bases for ideals of the ring of polynomials over the integers.