Igor Rochev
We prove, in a quantitative form, linear independence results for values of a certain class of q-series, which generalize classical q-hypergeometric series. These results refine our recent estimates.
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We prove linear independence results for values of (a certain class of) q-hypergeometric series in a quantitative form.
Algebraic independence of infinite series
本文给出了有理数级数集合代数独立的充分条件，并由此推导出级数无理性与有理数线性独立的新结果。
On algebraic independence of Taylor coefficients of certain Anderson-Thakur series
本文研究Anderson-Thakur级数泰勒系数的代数无关性问题，通过考虑与正特征多重zeta值相关的t-动机的延拓，在假设下确定t-动机伽罗瓦群维数，结合Papanikolas理论得到代数无关性结果。
The paper studies algebraic independence of certain reciprocal sums of Fibonacci and Lucas sequences. Also more general binary recurrences are considered. The main tool is Mahler's method reducing the investigation of the algebraic independence of function values to the one of functions if these satisfy certain types of functional equations.
A lower bound for the dimension of the $\Q$-vector space spanned by special values of a Dirichlet series with periodic coefficients is given. As a corollary, it is deduced that both special values at even integers and at odd integers contain infinitely many irrational numbers. This result is proved by T.Rivoal if the function considered is the Riemann zeta function, and this paper gives its generalization to more general Dirichlet series.
Linear independence of theta series of positive-definite even unimodular lattices
研究正定单模偶格中theta级数的线性独立性，确定其最小度数范围。
In this note we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue-Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method.
The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.
Let $\{F_{n}\}_{n\geq0}$ be the sequence of the Fibonacci numbers. The aim of this paper is to give explicit formulae for the infinite products \[ \prod_{n=1}^{\infty}\left( 1+\frac{1}{F_{n}}\right) ,\qquad\prod_{n=3}^{\infty}\left( 1-\frac{1}{F_{n}}\right) \] in terms of the values of the Jacobi theta functions. From this we deduce the algebraic independence over $\mathbb{Q}$ of the above numbers by applying Bertrand's theorem on the algebraic independence of the values of the Jacobi theta functions.
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a new technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Section 2 and 3, using the technique together with Mahler's method, we can reduce the al…
Linear independence of values of hypergeometric functions and arithmetic Gevrey series
证明广义超几何函数${}_pF_q$在不同代数点处值的线性独立性，利用II型帕德逼近统一构造及新型非零论证，推广加强前人成果。
Inspired by Warren Sinnott 's method we prove a linear independence result modulo p for the Iwasawa power series associated to Kubota-Leopoldt p-adic L-functions.
In this paper, we study multizeta values over function fields in characteristic $p$. For each $d \geq 2$, we show that when the constant field has cardinality $> 2$, the field generated by all multizeta values of depth $d$ is of infinite transcendence degree over the field generated by all single zeta values. As a special case, this gives an affirmative answer to the function field analogue of a question of Y.\ André.
In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated by a linear recurrence. In order to prove this result, we reduce the algebraic independency to that of Mahler functions of several variables by shifting the linear recurrence and apply the theory of Mahler functions.
Arithmetical independence of certain uniform sets of algebraic integers
本研究针对四组次数≤3的代数整数集合，证明其具备几乎一致性与算术独立性特征，并明确了这些集合生成的代数数域对二次、三次域的覆盖程度。
We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their $q$-analogs. Using this, we prove a propagation phenomenon: when these generating series are algebraically independent, this is also the case for their $q$-analogs.
A linear independence criterion for certain infinite series with polynomial orders
本文研究了当 $q$ 为 Pisot 或 Salem 数时，特定无穷级数在域 $\mathbb{Q}(q)$ 上的线性无关性，并针对 $a_j(n)$ 为多项式的情形提出了线性无关性判据。
An extension of algebraic independence of special values for non-lacunary power series
本文针对非稀疏幂级数在Pisot或Salem数倒数处的特殊值，提出了三个值代数独立性的新判据，并应用于证明特定级数和的代数独立性，且系数与指数选择灵活。
Here we propose a survey on Mahler's theory for transcendence and algebraic independence focusing on certain applications to the arithmetic of periods of Anderson t-motives.
We give some new results on algebraic independence within Mahler's method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In particular, our results furnishes, for $n\geq 1$ arbitrarily large, new examples of sets $(θ_1,...,θ_n)\in\mrr^n$ normal in the sense of definition formulated by Grigory Chudnovsky (1980).
We provide an alternative proof that the finite rational linear combination of radicals, under certain constraint, are linearly independent over $\mathbb{Q}$.
Let $(i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}$ and $S_{i,j}$ be an infinite subset of positive integers including all prime numbers in some arithmetic progression. In this paper, we prove the linear independence over $\mathbb{Q}$ of the numbers \[ 1, \quad \sum_{n\in S_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}, \] where $b\geq2$ is an integer and $a_{i,j}(n)$ are bounded nonzero integer-valued functions on $S_{i,j}$. Moreover, we also establish a necessary and sufficient condition on the subset $\mathcal{A}$ of $\mathbb{N}\times \mathbb{N}_{\ge…
We show that certain ramification invariants associated to a compatible system of $\ell$-adic sheaves on a curve are independent of $\ell$.
Linear independence of values of the $q$-exponential and related functions
本文证明当$q$为Pisot-Vijayraghavan数时，$q$-指数函数及其导数在代数点上的值线性无关，并推广至相关函数。
We prove linear independence of indefinite iterated Eisenstein integrals over the fraction field of the ring of formal power series $\mathbb{Z}[[q]]$. Our proof relies on a general criterium for linear independence of iterated integrals, which has been established by Deneufch{â}tel, Duchamp, Minh and Solomon. As a corollary, we obtain $\mathbb{C}$-linear independence of indefinite iterated Eisenstein integrals, which has applications to the study of elliptic multiple zeta values, as defined by Enriquez.
Rodrigues formula and linear independence for values of hypergeometric functions with parameters vary
本文通过证明广义 Rodrigues 公式，构造含 Gauss 超几何函数在内的 Padé 逼近，从而建立参数可变时其函数值在数域上的新线性无关性准则。
In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classe…
Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements
本文定义超图的标记多着色、标记色多项式与独立级数，证明其系数关系并推广至子空间排列，同时给出超平面排列的非负性结果及超图边基数猜想。
A linear independence criterion for $p$-adic Hurwitz zeta values at odd integers
该研究推广F. Beukers的定理，建立p进Hurwitz zeta函数在奇数s≥3处取值的线性独立性新准则，通过构造满足f(z)=f(1-z)的Laurent级数的Padé逼近完成证明，补充了p进多伽马值线性独立性的相关工作。
On the characterization of chordal graphs using Horn hypergeometric series
本文使用初等组合方法，基于多变量独立多项式的逆与多色色多项式的联系，重新证明了弦图可由独立多项式的逆为 Horn 超几何级数来刻画的结论。
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…
We prove a result which describes, for each $n\ge 1$, all linear dependencies among $n$ images in elliptic curves of special points in modular or Shimura curves under parameterizations (or correspondences). Our result unifies and improves in certain aspects previous work of Rosen-Silverman--Kühne and Buium-Poonen.
We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasi-projective variety over $\overline{\mathbf{Q}}$ that are integral curves of some algebraic vector field (defined over $\overline{\mathbf{Q}}$). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields. This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series $E_2,E_4,E_6$. The main technical improvement in our a…
Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, e.g., Zeilberger's algorithm fails to find a recurrence with minimal order.
Linear independence measures for Chowla--Selberg periods
通过同时Padé近似估计超几何函数，研究特定d和D下1、π√d、Ω_D/π和π/Ω_D的线性形式，揭示其线性独立性。
Linear independence of series related to the Thue--Morse sequence along powers
本文推广E. Miyanohara的结果，证明当β为大于√φ的Pisot数或Salem数时，与沿幂次的Thue–Morse序列相关的级数在ℚ(β)上线性无关，进而推出对应线性组合序列非最终周期。
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