Arithmetic and Geometry pp 61-105 | Cite as

# Number Theoretic Applications of Polynomials with Rational Coefficients Defined by Extremality Conditions

## Abstract

It is well known that classes of polynomials in one variable defined by various extremality conditions play an extremely important role in complex analysis. Among these classes we find orthogonal polynomials (especially classical orthogonal polynomials expressed as hypergeometric polynomials) and polynomials least deviating from zero on a given continuum (Chebicheff polynomials). Orthogonal polynomials of the first and second kind appear as denominators and numerators of the Padé approximations to functions of classical analysis and satisfy familiar three-term linear recurrences. These polynomials were used repeatedly to study diophantine approximations of values of functions of classical analysis, especially exponential and logarithmic functions at rational points *x* = *p/q* [1], [2], [3], [4], [5]. The methods of Padé approximation in diophantine approximations are quite powerful and convenient to use, since they replace the problem of rational approximations to numbers with the approximations of functions. There are, however, arithmetic restrictions on rational approximations to functions if they are to be used for diophantine approximations. The main restriction on polynomials here is to have rational integer coefficients or rational coefficients with a controllable denominator. Such arithmetic restrictions transform a typical problem of classical analysis into an unusual mixture of arithmetic and analytic difficulties. For example, recurrences defining orthogonal polynomials must be of a special type to guarantee that their solutions will have hounded denominators. In this paper we consider various classes of polynomials generated by imposing arithmetic restrictions on classical approximation theory problems (orthogonal or Chebicheff polynomials).

## Keywords

Orthogonal Polynomial Rational Approximation Singular Integral Equation Algebraic Number Monodromy Group## Preview

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## References

- [1]L. V. Ahlfors,
*Conformal*invariants, McGraw-Hill, 1973.Google Scholar - [2]A. O. Gelfond, L. G. Schnirelman, Uspekhi Math., Nauk, No. 2, (1936) (Russian).Google Scholar
- [3]A. O. Gelfond, Comment “On determining the number of primes not exceeding the given values” and “On primes” in P. L. Chebicheff,
*Complete Works*,*v.*1., Academy, Moscow, 1946, 2nd printing (Russian).Google Scholar - [4]Le Baron O. Ferguson,
*Approximation by polynomials with integral coefficients*, Mathematical surveys #17, American Mathematical Society, Providence, 1980.Google Scholar - [5]R. M. Trigub,
*Metric questions of the theory of functions and mappings*, No. 2., pp. 267–333, Naukova Dumka Publishing House, Kiev, 1971 ( Russian).Google Scholar - [6]A. E. Ingham,
*The distribution of prime numbers*, Cambridge Tracts, #30, Cambridge Univ. Press, 1932.Google Scholar - [7]M. Fekete, C. R. Acad. Sci. Paris. 1923, 17.Google Scholar
- [8]A. O. Celfond, Uspekhi Math. Nauk. 10 (1955), No. 1, 41–56 (Russian).Google Scholar
- [9]E. Aparicio, Revista Matematica Hispano-Americana, 4 Serie, t. XXXVIII, No. 6, 259–70 Madrid, 1978.Google Scholar
- [10]E. Aparicio, Revista de la Universidad de Santader, Numero 2, Parte 1 (1979), 289–91.Google Scholar
- [11]E. Aparicio, Revista Matematica Hispano-Americana, 4 Serie, v. XXXVI (1976), 105–24.Google Scholar
- [12]A. Selberg, Norsk Mathematisk Tidsskrift, 26 (1944), 71–78.MathSciNetMATHGoogle Scholar
- [13]G. Szegö,
*Orthogonal polynomials*, American Mathematical Society Colloquium Publication #23, Providence, Rhode Island, 1939.Google Scholar - [14]L. Hulthén, Arkiv for Mat. Astron. Fysik, 26A, Hafte 3, No. 11 (1938), 1–106.Google Scholar
- [15]D. V. Chudnovsky, G. V. Chudnovsky, Letters Nuovo Cimento, 19, No. 8 (1977), 300–02.CrossRefGoogle Scholar
- [16]J. Nuttall,
*Bifurcation phenomena in mathematical physics and related topics*,D. Reidel Publishing Company, Boston, 1980, 185202.Google Scholar - [17]N. I. Muskhelishvili,
*Singular integral equations*, P. Noordhoff, N. V. Groningen, Holland, 1953.Google Scholar - [18]
- [19]J. J. Sylvester, Messenger. Math. (2), 21 (1891), p. 120.Google Scholar
- [20]E. Landau, Sitz. Akad. Wissen. Wien, Math-Nat. Klasse, bd. CXVII, Abt. Ha, 1908.Google Scholar
- [21]M. Nair, The American Mathematical Monthly, 89, No. 2, (1982), 126–29.MathSciNetMATHCrossRefGoogle Scholar
- [22]M. Nair, J. London Math. Soc. (2), 25 (1982), 385–91.MathSciNetMATHCrossRefGoogle Scholar
- [23]H. G. Diamond, K. S. McCurley, Lecture Notes Math., v. 899, Springer, 1981, 239–53.Google Scholar
- [24]D. G. Cantor, J. Reine Angew. Math., 316 (1980), 160–207.MathSciNetMATHGoogle Scholar