Nesterenko's linear independence criterion for vectors
Stéphane Fischler
备注：
22 pages. With respect to the first version, the main result has been generalized; the application to zeta values is now much stronger, and its proof is postponed to another paper
In this paper we deduce a lower bound for the rank of a family of $p$ vectors in $\R^k$ (considered as a vector space over the rationals) from the existence of a sequence of linear forms on $\R^p$, with integer coefficients, which are small at $k$ points. This is a generalization to vectors of Nesterenko's linear independence criterion (which corresponds to $k=1$), used by Ball-Rivoal to prove that infinitely many values of Riemann zeta function at odd integers are irrational. The proof is based on geometry of numbers, namely Minkowski's theorem on convex bodies.
We consider the problem of simultaneous approximation of real numbers $θ_1, \ldots,θ_n$ with rationals and the dual problem of approximating zero with the values of the linear form $x_0+θ_1x_1+\ldots+θ_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation, which proves their result. We also derive several corollaries previously unknown. Particularly, we show that, together with the transference inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalit…
Let $m\ge 2$ be an integer, $K$ an algebraic number field and $α\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in $1,{\rm{log}}(1+α),\ldots,{\rm{log}}^{m-1}(1+α)$ with algebraic integer coefficients in both complex and $p$-adic cases (see Theorem $2.1$ and Theorem $2.4$). Especially, in the complex case, our result is a refinement of the result of Nesterenko-Waldschmidt on the lower bound of linear form in certain values of power of logarithms. The main integrant is based on Hermite-Mahler's Padé approximation of e…
Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For any fixed algebraic number $α$ such that $0 \textless{} \vert α\vert \textless{} R$ and any number field $\mathbb{K}$ containing $α$ and the $A\_k$'s, we define $Φ\_{α, S}$ as the $\mathbb{K}$-vector space generated by the values $F\_n^{[s]}(α)$, $n\ge 1$ and $0\leq s\leq S$. We prove that $u\_{\mathbb{K},F}\log(S)\leq \dim\_{\mathbb{K}}(Φ\_{α, S})\leq v\_F S$…
We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the $\mathbb{Q}$-vector space spanned by $1$ and those three numbers has dimension at least 3, whenever we are unable to achieve full linear independence, by using simultaneous approximations, i.e. those usually arising from Hermite-Padé approximations of type II and their suitable generalizations. It should be recalled that approximations of type I and II are rela…
In 1995, Thakur invented and studied positive characteristic analogues of hypergeometric functions. In this paper, we interpret the special values of those functions as periods of a pre-$t$-motive. As a consequence, we show their transcendence and linear independence results by using Chang's refined version of the Anderson-Brownawell-Papanikolas criterion. Furthermore, as by-products, we show some linear/algebraic independence results among the special values of Kochubei multiple polylogarithms according to our period interpretation and the corresponding refined criterion.
We develop a new method for proving algebraic independence of $G$-functions. Our approach rests on the following observation: $G$-functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with the Frobenius that are obtained by reduction modulo prime ideals. When these linear difference equations have order one, the coefficients of the $G$-function satisfy congruences reminiscent of a classical theorem of Lucas on binomial coefficients. We use this to derive a Kolchin-like algebraic independence…
Let $θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$ with $q=e^{iπτ}$ denote the Thetanullwert of the Jacobi theta function \[θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.\] Moreover, let $θ_2(τ)=2\sum_{ν=0}^{\infty} q^{{(ν+1/2)}^2}$ and $θ_4(τ)=1+2\sum_{ν=1}^{\infty} {(-1)}^νq^{ν^2}$. For algebraic numbers $q$ with $0<|q|<1$ and for any $j\in \{ 2,3,4\}$ we prove the algebraic independence over $\mathbb{Q}$ of the numbers $θ_j(nτ)$ and $θ_j(τ)$ for all odd integers $n\geq 3$. Assuming the same conditions on $q$ and $τ$ as above, we obtain sufficient conditions by use of a criterion invol…
For a given rational number $x$ and an integer $s\geq 1$, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by $$Φ_{s}(x,z)= \sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\enspace.$$ We prove the linear independence over any number field $K$ of the numbers $1$ and $Φ_{s_j}(x_j,α_i)$ with any choice of distinct shifts $x_1,\ldots, x_d$ with $0\le x_1<\ldots<x_d<1$, as well as any choice of depths $1\leq s_1\leq r_1,\ldots, 1\leq s_d\leq r_d$, at distinct algebraic numbers $α_1,\ldots,α_m\in K$ subject to a metric condition. As is usual in the theory,…
