Related papers: Mahler's method and Carlitz logarithm

In the restricted setting of product phase space lattices, we give an alternate proof of P. Linnell's theorem on the finite linear independence of lattice Gabor systems in $L^2(\mathbb R^d)$. Our proof is based on a simple argument from the…

We show that $T_p(z)=\prod_{j=1}^{\infty}(1-z^{p^{j}})^{-1/p^{j}}$ is transcendental over $\overline{\mathbb{Q}}(z)$, and establish the transcendence of its values at nonzero algebraic points inside the unit disk. Furthermore, we obtain an…

Let $K$ be a finite extension of $\mathbb{Q}_p$ that is totally ramified over $\mathbb{Q}_p$. The set $\mathcal{M}\mathcal{F}(K)$ consists of power series in $1+zK[[z]]$ that are solutions of differential operators in $K(z)[d/dz]$ equipped…

For finite extensions of a rational function field over a finite field, we prove a "P-adic class formula" in the spirit Taelman's work.

For a lattice \Lambda in the complex plane, let K_{\Lambda} be the field of \Lambda-elliptic functions. For two relatively prime integers p (respectively q) greater than 1, consider the endomorphisms \psi (resp. \phi) of K_{\Lambda} given…

Working in point-free topology under the constraints of geometric logic, we prove the Fundamental Theorem of Calculus, and apply it to prove the usual rules for the derivatives of $x^\alpha$, $\gamma^x$, and $\log_\gamma x$.

A method is developed to construct a non-local massless scalar field theory in a flat quantised space-time generated by an operator algebra. Implicit in the operator algebra is a fundamental length scale of the space-time. The fundamental…

We define two pairings relating the A-motive with the dual A-motive of an abelian Anderson A-module. We show that specializations of these pairings give the exponential and logarithm functions of this Anderson A-module, and we use these…

A complete set of supertraces on the algebras of observables of the rational Calogero models with harmonic interaction based on the classical root systems of B_N, C_N and D_N types is found. These results extend the results known for the…

Anderson t-modules are analogs of abelian varieties in positive characteristic. Associated to such a t-module, there are its t-motive and its dual t-motive. When dealing with these objects, several questions occur which one would like to…

Although it is easy to prove the sufficient conditions for optimality of a linear program, the necessary conditions pose a pedagogical challenge. A widespread practice in deriving the necessary conditions is to invoke Farkas' lemma, but…

Linear forms in logarithms over connected commutative algebraic groups over the algebraic numbers field have been studied widely. However, the theory of linear forms in logarithms over noncommutative algebraic groups have not been developed…

We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric function field Gamma-function are explained…

We explore Mahler numbers originating from functions $f(z)$ that satisfy the functional equation $f(z) = (A(z)f(z^d) + C(z))/B(z)$. A procedure to compute the irrationality exponents of such numbers is developed using continued fractions…

We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a…

Let K be a field of characteristic 0 and A be a rigid tensor K-linear category. Let M be a finite-dimensional object of A in the sense of Kimura-O'Sullivan. We prove that the "motivic" zeta function of M with coefficients in K\_0(A) has a…

Using the same heuristic argument leading to the Lang-Waldschmidt Conjecture in the theory of linear forms in logarithms, we formulate an effective version of the Linear Independence conjecture for the ordinates of the non-trivial zeros of…

Let $\mathbb{K}$ be a function field of characteristic $p>0$. We recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over $\mathbb{K}(z)$. This paper is dedicated to proving the following…

We consider Mahler functions $f(z)$ which solve the functional equation $f(z) = \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and $d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either $f(b)$ is…

We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only…