Оценки многочленов от значений Е-функций

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

Let G be a commutative algebraic group embedded in projective space and \(\Gamma \) a finitely generated subgroup of G. From these data we construct a chain of algebraic subgroups of G which is intimately related to obstructions to multiplicity or interpolation estimates used in transcendental number theory and algebraic independence. Let \(\gamma _1,\ldots ,\gamma _l\) denote a family of generators of \(\Gamma \) and, for any \(S>1\), let \(\Gamma (S)\) be the set of elements \(n_1\gamma _1+\cdots +n_l\gamma _l\) with integers \(n_j\) such that \(|n_j| < S\). Then this chain of subgroups controls, for large values of S, the distribution of \(\Gamma (S)\) with respect to algebraic subgroups of G. As an application we essentially determine (up to multiplicative constants) the locus of common zeros of all \(P \in H^0(\overline{G} ,\mathcal{O}(D))\) which vanish to at least some given order at all points of \(\Gamma (S)\). When D is very small this result reduces to a multiplicity estimate; when D is very large it is a kind of interpolation estimate.

This paper explores a novel application of the Laplace transform in cryptographic schemes by employing a linear combination of Bessel functions and exponential functions. By leveraging the transform’s properties, we develop an encryption-decryption framework that is computationally efficient and highly secure against certain types of cryptographic attacks. The exponential decay observed in Laplace-transformed domains and the complex behaviour of Bessel functions are powerful tools for data encoding and concealment. Analytical and numerical results validate the method’s effectiveness, showcasing its potential for modern cryptographic systems. Cryptography has long been a cornerstone of secure communication, with its methods evolving in tandem with technological advancements. While traditional schemes rely on number theory or algebraic structures, this paper explores an alternative approach that combines Laplace transforms with special functions. Specifically, we focus on employing Bessel functions in conjunction with exponential functions to construct an encryption algorithm. The Laplace transform’s unique properties, such as linearity, time-shifting, and frequency domain representation, offer a fertile ground for developing innovative cryptographic methods. Then the algorithm described in this study generates a dynamic cipher by processing plaintext using polynomial, exponential, and factorial-based operations. The four fundamental steps comprising the computational process are text preprocessing, dynamic coefficient calculation, encryption packaging, and reverse analysis. The computational cost of each step varies depending on the number of characters processed. The symbol for this character quantity is n. The cryptographic analysis of the problem is discussed in detail.

This volume consists of a collection of papers devoted primarily to transcendental number theory and diophantine approximations written by the author. Most of the materials included in this volume are English translations of the author's Russian manuscripts, extensively rewritten and brought entirely up to date. These papers and other papers included in this volume were available to specialists in manuscript form, but this is the first time that they have been collected and published.Though the earlier papers have been preserved in the form in which they were prepared initially, the volume is organized in such a way as to reflect recent progress and to allow readers to follow recent developments in the field. As an introductory guide to the volume, the author included an expanded and updated text of his invited address on his work on the theory of transcendental numbers to the 1978 International Congress of Mathematicians in Helsinki. The appendix contains a paper on the extremality of certain multidimensional manifolds prepared by A. I. Vinogradov and the author in 1976. Chudnovsky received a MacArthur Foundation Fellowship in 1981.

The article takes a look at transcendence and algebraic independence problems, introduces statements and proofs of theorems for some kinds of elements from direct product of 𝑝-adic fields and polynomial estimation theorem. Let Q𝑝 be the 𝑝-adic completion of Q, Ω𝑝 be the completion of the algebraic closure of Q𝑝, 𝑔 = 𝑝1𝑝2 . . . 𝑝𝑛 be a composition of separate prime numbers, Q𝑔 be the 𝑔-adic completion of Q, in other words Q𝑝1 ⊕. . .⊕Q𝑝𝑛. The ring Ω𝑔 ∼=Ω𝑝1⊕...⊕Ω𝑝𝑛, a subring Q𝑔, transcendence and algebraic independence over Q𝑔 are under consideration. Also, hypergeometric series $$𝑓(𝑧) =∞Σ𝑗=0((𝛾1)𝑗 . . . (𝛾𝑟)𝑗)/((𝛽1)𝑗 . . . (𝛽𝑠)𝑗)(𝑧𝑡)^𝑡𝑗 $$, and their formal derivatives are under consideration. Sufficient conditions are obtained under which the values of the series 𝑓(𝛼) and formal derivatives satisfy global relation of algebraic independence, if 𝛼 =∞Σ𝑗=0 𝑎_𝑗𝑔^(𝑟_𝑗), where 𝑎𝑗 ∈ Z𝑔, and non-negative rationals 𝑟𝑗 increase strictly unbounded.

In this chapter, we will examine some of the emerging themes in the theory of transcendental numbers. The most fascinating is the “modular connection” linking it with the theory of modular forms. We have met a part of this connection in the earlier chapters. In this chapter, we will indicate some other relations.

Siegel’s Lemma is concerned with finding a “small” nontrivial integer solution of a large system of homogeneous linear equations with integer coefficients, where the number of variables substantially exceeds the number of equations (for example, n equations and N variables with N ≥ 2n), and “small” means small in the maximum norm. Siegel’s Lemma is a clever application of the Pigeonhole Principle, and it is a pure existence argument. The basically combinatorial Siegel’s Lemma is a key tool in transcendental number theory and diophantine approximation. David Masser (a leading expert in transcendental number theory) asked the question whether or not the Siegel’s Lemma is best possible. Here we prove that the so-called “Third Version of Siegel’s Lemma” is best possible apart from an absolute constant factor. In other words, we show that no other argument can beat the Pigeonhole Principle proof of Siegel’s Lemma (apart from an absolute constant factor). To prove this, we combine a concentration inequality (i.e., Fourier analysis) with combinatorics.

Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K ⊆ ℝ is a field of finite transcendence degree over the rationals, then every weakly o-minimal expansion of (K,≤,+,·) is polynomially bounded. In the special case where K is the field of all real algebraic numbers, we give a proof which makes use of a much weaker result from transcendental number theory, namely, the Gelfond-Schneider theorem. Apart from this we make a couple of observations concerning weakly o-minimal expansions of arbitrary ordered fields of finite transcendence degree over the rationals. The strongest result we prove says that if K is a field of finite transcendence degree over the rationals, then all weakly o-minimal non-valuational expansions of (K,≤,+,·) are power bounded.

Integer factoring is a curious number theory problem with wide applications in complexity and cryptography. The best known algorithm to factor a number n takes time, roughly, exp(2*log^{1/3}(n)*log^{2/3}(log(n))) (number field sieve, 1989). One basic idea used is to find two squares, possibly in a number field, that are congruent modulo n. Several variants of this idea have been utilized to get other factoring algorithms in the last century. In this work we intend to explore new ideas towards integer factoring. In particular, we adapt the AKS primality test (2004) ideas for integer factoring. In the motivating case of semiprimes n=pq, i.e. p<q are primes, we exploit the difference in the two Frobenius morphisms (one over F_p and the other over F_q) to factor n in special cases. Specifically, our algorithm is polynomial time (on number theoretic conjectures) if we know a small algebraic dependence between p,q. We discuss families of n where our algorithm is significantly faster than the algorithms based on known techniques.

The first remark that must be made about integration in finite terms is that all the algorithms, and nearly all the implementations (Wang [1971] is the exception), deal with indefinite integration. Definite integration is largely restricted to a combination of table look-up and "guess the contour" heuristics. As has often been pointed out [Askey, 1984], definite integration is far more applicable than indefinite integration, since many special functions (β and Γ functions, for example) are defined by definite integrals, and whole tools, such as Laplace transforms, are built on definite integrals. We note that these problems are fundamentally algebraic, rather than numerical, since we are computing, not a number, but a function defined by F ( y ) = ∫ f ( x , y )d x . Very little is known about algorithmic definite integration, though recent work in transcendence theory (see, e.g., Chudnovsky [1982]) may at least provide some methods for showing negative results.

Chapter 4 - Special functions

Finite fields analogs of the classical error function and incomplete gamma function are defined as complex functions over finite fields.

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.

Many results in transcendental number theory are proved using the following principle. Suppose that we want to prove that a number α is irrational or transcendental. To do this, we would like to find a way to construct a sequence of sufficiently good Diophantine approximations to α — perhaps approximations by rational numbers (see §1.4 below), or else a sequence of simultaneous approximations (see §2), or possibly a sequence of polynomials with integer coefficients that take small values at α (see §4.2). If α were rational or algebraic, there would be some polynomial P(x) ∈ ℤ[x], P ≢ 0, having α as a root. If our sequence of approximations is “dense” enough, then we can use it to eliminate α in the equation P(α) = 0, thereby obtaining a contradiction.KeywordsLinear FormRational NumberHypergeometric FunctionAlgebraic NumberContinue Fraction ExpansionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We construct asymptotic expansions for the normalised incomplete gamma function Q ( a , z ) = Γ ( a , z ) / Γ ( a ) Q(a,z)=\Gamma (a,z)/\Gamma (a) that are valid in the transition regions, including the case z ≈ a z\approx a , and have simple polynomial coefficients. For Bessel functions, these types of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as a → − ∞ a\to -\infty of the unique negative zero of the regularised incomplete gamma function γ ∗ ( a , x ) \gamma ^*(a,x) .