Localization of the complex zeros of parametrized families of polynomials

AbstractLet Pn(x)=xm+pm−1(n)xm−1+⋯+p1(n)x+pm(n) be a parametrized family of polynomials of a given degree with complex coefficients pk(n) depending on a parameter n∈Z≥0. We use Rouché’s theorem to obtain approximations to the complex roots of Pn(x). As an example, we obtain approximations to the complex roots of the quintic polynomials Pn(x)=x5+nx4−(2n+1)x3+(n+2)x2−2x+1 studied by A. M. Schöpp

Simple zeros of Dedekind zeta functions

International audienceUsing Stechkin's lemma we derive explicit regions of the half complex plane R (s) = 1 - c = logd(K) and vertical bar gamma vertical bar <= d/logd(K) for some absolute positive constants c and d. These regions are larger and our proof is simpler than recently published such regions and proofs

Fundamental units for orders of unit rank 1 and generated by a unit

Banach Center PublicationsInternational audienceLet ε be an algebraic unit for which the rank of the group of units of the order ℤ[ε] is equal to 1. Assume that ε is not a complex root of unity. It is natural to wonder whether ε is a fundamental unit of this order. It turns out that the answer is in general yes, and that a fundamental unit of this order can be explicitly given (as an explicit polynomial in ε) in the rare cases when the answer is no. This paper is a self-contained exposition of the solution to this problem, solution which was up to now scattered in many papers in the literature

Explicit upper bounds for values at s=1 of Dirichlet L-series associated with primitive even characters

AbstractLet S be a given finite set of pairwise distinct rational primes. We give an explicit constant κS such that for any even primitive Dirichlet character χ of conductor qχ>1 we have∏p∈S1−χ(p)pL(1,χ)⩽12∏p∈S1−1p(logqχ+κS)+o(1),where o(1) is an explicit error term which tends rapidly to zero when qχ goes to infinity

Discriminants of Sn\mathop{\mathfrak{S}}\nolimits_n-orders

International audienceLet alpha be an algebraic integer of degree n >= 2. Let alpha(1),..., alpha(n) be the n complex conjugate of alpha. Assume that the Galois group Gal(Q(alpha(1),..., alpha(n))/Q) is isomorphic to the symmetric group S-n. We give a Z-basis and the discriminant of the order Z[alpha(1),..., alpha(n)]. We end up with an open question showing that this problem seems much harder when we assume that Q(alpha)/Q is already Galois or even cyclic

When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

summary:Let $\varepsilon$ be an algebraic unit of the degree $n\geq 3$. Assume that the extension ${\mathbb Q}(\varepsilon )/{\mathbb Q}$ is Galois. We would like to determine when the order ${\mathbb Z}[\varepsilon ]$ of ${\mathbb Q}(\varepsilon )$ is ${\rm Gal}({\mathbb Q}(\varepsilon )/{\mathbb Q})$-invariant, i.e. when the $n$ complex conjugates $\varepsilon _1,\cdots ,\varepsilon _n$ of $\varepsilon$ are in ${\mathbb Z}[\varepsilon ]$, which amounts to asking that ${\mathbb Z}[\varepsilon _1,\cdots ,\varepsilon _n]={\mathbb Z}[\varepsilon ]$, i.e., that these two orders of ${\mathbb Q}(\varepsilon )$ have the same discriminant. This problem has been solved only for $n=3$ by using an explicit formula for the discriminant of the order ${\mathbb Z}[\varepsilon _1,\varepsilon _2,\varepsilon _3]$. However, there is no known similar formula for $n>3$. In the present paper, we put forward and motivate three conjectures for the solution to this determination for $n=4$ (two possible Galois groups) and $n=5$ (one possible Galois group). In particular, we conjecture that there are only finitely many cyclic quartic and quintic Galois-invariant orders generated by an algebraic unit. As a consequence of our work, we found a parametrized family of monic quartic polynomials in ${\mathbb Z}[X]$ whose roots $\varepsilon$ generate bicyclic biquadratic extensions ${\mathbb Q}(\varepsilon )/{\mathbb Q}$ for which the order ${\mathbb Z}[\varepsilon ]$ is ${\rm Gal}({\mathbb Q}(\varepsilon )/{\mathbb Q})$-invariant and for which a system of fundamental units of ${\mathbb Z}[\varepsilon ]$ is known. According to the present work it should be difficult to find other similar families than this one and the family of the simplest cubic fields

Upper bounds on L (1, χ ) taking into account ramified prime ideals

International audienceLet chi range over the non-trivial primitive characters associated with the abelian extensions L/K of a given number field K, i.e. over the non-trivial primitive characters on ray class groups of K. Let f(chi) be the norm of the finite part of the conductor of such a character. It is known that vertical bar L(1, chi)I <= 1/2-Res(s=1)(xi(K)(s)) log f(chi) + O(1), where the implied constants in this 0(1) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of chi, i.e. ramified prime ideals. This has already been done only in the case that K = Q. This paper is devoted to giving for the first time such improvements for any K. As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field