Zakład Teorii Liczb

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Research during the last few years concerned different branches of number theory as well as fields and polynomials, which will be reviewed in order adopted by Mathematical Reviewers.

In elementary number theory a lower estimate has been proved under certain conditions in [9] for the number of solutions of a linear homogeneous congruence in a multidimensional box. The same, best possible, estimate under different conditions is in the course of publications.

A problem on diophantine equations over rational integers has been solved in [1], binary forms over an arbitrary field have been considered in [7] and forms in many variables in [14]. An almost explicit construction of a point on an elliptic curve over a finite field has been given in [17].

A problem on the length (a kind of a height) of polynomials with real coefficients, related to diophantine approximation has been studied in [8] and [12], a similar study of polynomials with complex coefficients is in course of publication. Paper [6] studies a problem in geometry of numbers.

Elementary analytic number theory is represented by [16] and a more advanced on by [5]. [15] concerns elementary algebraic number theory [2] K-theory and [3] finite fields.

In field theory and polynomials papers [3] and [11] are concerned with localization of zeros of polynomials in one variable, papers [10] and [11] with reducibility of symmetric polynomials.

The following topics have been studied in the period 1999--2008:

  1. Representation of integer vectors as a linear combination of shorter integer vectors (I. Aliev, A. Schinzel)
  2. The Milnor group K2F (J. Browkin)
  3. Distribution of primitive roots (A. Paszkiewicz, A. Schinzel)
  4. Pseudoprimes and their generalizations (A. Rotkiewicz, A. Schinzel)
  5. Reducibility of polynomials (A. Schinzel)
  6. The number of non-zero coefficients of the greatest common divisor of two polynomials with given numbers of non-zero coefficients (A. Schinzel)
  7. The Mahler measure and other measures of polynomials (A. Schinzel)
  8. Weak automorphs of binary forms (A. Schinzel)
  9. Number of solutions of a linear homogeneous congruence in a box (A. Schinzel)
  10. Representation of a multivariate polynomial as a sum of univariate polynomials in linear forms (A. Schinzel)
  11. Polynomial and exponential congruences to a prime modulus (M. Skałba, A. Schinzel)
  12. Congruences for L-functions (J. Urbanowicz)
  13. Divisibility of a generalized Vandermonde determinant by powers of two (J. Urbanowicz)

Several people not employed by the Number Theory Section have collaborated in the study of the above topics, namely W. Schmidt in 1), M. Zakarczemny in 9), A. Białynicki in 10), K. Williams in 12) and S. Spież in 13).

Research papers published in 2005--2008 (March)

  1. J. Browkin (with J. Brzeziński), On sequences of squares with constant second differences, Canadian Math. Bulletin 48 (2006), 481--491.
  2. J. Browkin, Elements of small order in K2F, II, Chin. Ann. Math. Ser. B 28 (2007), 507--520.
  3. A. Schinzel, Self-inversive polynomials with all zeros on the unit circle, Ramanujan Journal 9 (2005), 19--23.
  4. A. Schinzel (with T. Bolis), Identities which imply that a ring is Boolean, Bull. Greek Math. Soc. 48 (2003), 1--5 (antedated).
  5. A. Schinzel (with S. Kanemitsu and Y. Tanigawa), Sums involving the Hurwitz zeta-function values, Zeta Function, Topology and Quanture Physics, 81-90, Springer 2005.
  6. A. Schinzel (with I. Aliev and W. M. Schmidt), On vectors whose span contains a given linear subspace, Monatsh. Math. 144 (2005), 177-191.
  7. A. Schinzel, On weak automorphs of binary forms over an arbitrary field, Dissert. Math. 434 (2005), 48 pp.
  8. A. Schinzel, On the reduced length of a polynomial, Functiones et Approximatio 35 (2006), 271--306.
  9. A. Schinzel (with M. Zakarczemny), On a linear homogeneous congruence, Colloq. Math. 106 (2006), 283--292.
  10. A. Schinzel, Reducibility of symmetric polynomials, Bull. Polish Acad. Sci. Mathematics 53 (2005), 251--258 (antedated).
  11. A. Schinzel (with L. Losonczi), Self-inverse polynomials of odd degree, Ramanujan Journal 14 (2007), 305--320.
  12. A. Schinzel, On the reduced length of a polynomial with real coefficients, II, Functiones et Approximatio 37 (2007), 445--459.
  13. A. Schinzel, Reducibility of a special symmetric form, Acta Math. Universitatis Ostraviensis 14 (2006), 71--74 (antedated).
  14. A. Schinzel (with A. Białynicki-Birula), Representations of multivariate polynomials by sums of univariate polynomials in linear forms, Colloq. Math. 112 (2008), 201--233.
  15. M. Skałba, On sets which contain a q-th power residue for almost all prime modules, Colloq. Math. 102 (2005), 67--71.
  16. M. Skałba, Primes dividing both 2n and 3n-2 are rare, Arch. Math. 84 (2005), 485--495.
  17. M. Skałba, Points on elliptic curves over finite fields, Acta Arith. 117 (2005), 293--301.

Besides the following book has been published
A. Schinzel, Selecta (2 vols), ed. H. Iwaniec, W. Narkiewicz, J. Urbanowicz, Zürich 2007.