Related papers: Des crit\`eres de transcendance inspir\'es par un …
Transcendence criteria inspired by Kolberg's paper dated 1962. This is the second part of a note about Kolberg's proof that the values of the sums of a class of certain power series in x, for algebraic values of x, are transcendent. A…
Using an application of Schmidt's Subspace Theorem, this paper gives new transcendence criteria for rapidly converging infinite products of algebraic numbers. The paper also improves existing criteria for irrationality of products and…
In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a series of positive rational terms is a transcendental number.…
Using Schmidt's Subspace Theorem, this paper improves and extends an existing transcendence result for sequences of algebraic numbers. The theorems thus produced correspond to a central theorem on the irrationality of sequences due to…
We review Hilbert's classical analytical proof of the transcendence of the number $\mathrm{e}$. Then, we show how this result can be obtained algebraically by means of formal power series (FPS). We give two proofs of the transcendence of…
For a finitely generated algebra over a field, the transcendence degree is known to be equal to the Krull dimension. The aim of this paper is to generalize this result to algebras over rings. A new definition of the transcendence degree of…
In this paper we investigate the infinite convergent sum $T=\sum_{n=0}^\infty\frac{P(n)}{Q(n)}$, where $P(x)\in\bar{\mathbb{Q}}[x]$, $Q(x)\in\mathbb{Q}[x]$ and $Q(x)$ has only simple rational zeros. N. Saradha and R. Tijdeman have obtained…
In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a sum, product and quotient of some series of positive rational…
We prove a functional transcendence theorem for the integrals of algebraic forms in families of algebraic varieties. This allows us to prove a geometric version of Andr\'e's generalization of the Grothendieck period conjecture, which we…
E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like…
Recently in Reference [ quant-ph/0202121] a computational criterion of separability induced by greatest cross norm is proposed by Rudolph. There, Rudolph conjectured that the new criterion is not weaker than positive partial transpose…
It is a well-known result that, after adding one Cohen real, the transcendence degree of the reals over the ground-model reals is continuum. We extend this result for a set $X$ of finitely many Cohen reals, by showing that, in the forcing…
Classical results on Diophantine approximation, such as Roth's theorem, provide the most effective techniques for proving the transcendence of special kinds of continued fractions. Multidimensional continued fractions are a generalization…
Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $\delta\ge 1$ and usual height $H\ge 1$, and let $\alpha\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^\alpha$: for any…
A singular point on a plane conic defined over $\mathbb{Q}$ is a transcendental point of the curve which admits very good rational approximations, uniformly in terms of the height. Extremal numbers and Sturmian continued fractions are…
Let $b$ be an algebraic number with $|b|>1$ and $\mathcal{H}$ a finite set of algebraic numbers. We study the transcendence of numbers of the form $\sum_{n=0}^\infty \frac{a_n}{b^n}$ where $a_n \in \mathcal{H}$ for all $n\in\mathbb{N}$. We…
For a given transcendental number $\xi$ and for any polynomial $P(X)=: \lambda_0+\cdots+\lambda_k X^k \in \mathbb{Z}[X]$, we know that $ P(\xi) \neq 0.$ Let $k \geq 1$ and $\omega (k, H)$ be the infimum of the numbers $r > 0$ satisfying the…
Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2…
Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf{PA}$. This result is commonly known as Feferman's completeness theorem. The purpose of this…
In this survey, we present five different proofs for the transcendence of Kempner's number, defined by the infinite series $\sum_{n=0}^{\infty} \frac{1}{2^{2^n}}$. We take the opportunity to mention some interesting ideas and methods that…