Zakład Teorii Liczb
- prof. dr hab. Jerzy Browkin (prof. zw.)
pok. 102, tel.: 22 522 81 02
- prof. dr hab. Jerzy Kaczorowski (prof. zw.)
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
In elementary number theory a lower estimate has been proved under
certain conditions in  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
A problem on diophantine equations over rational integers has been
solved in , binary forms over an arbitrary field have been
considered in  and forms in many variables in . An almost
explicit construction of a point on an elliptic curve over a
finite field has been given in .
A problem on the length (a kind of a height) of polynomials with
real coefficients, related to diophantine approximation has been
studied in  and , a similar study of polynomials with
complex coefficients is in course of publication. Paper 
studies a problem in geometry of numbers.
Elementary analytic number theory is represented by  and a
more advanced on by .  concerns elementary algebraic number
theory  K-theory and  finite fields.
In field theory and polynomials papers  and  are concerned
with localization of zeros of polynomials in one variable,
papers  and  with reducibility of symmetric polynomials.
The following topics have been studied in the period 1999--2008:
- Representation of integer vectors as a linear
combination of shorter integer vectors (I. Aliev, A. Schinzel)
- The Milnor group K2F (J. Browkin)
- Distribution of primitive roots (A. Paszkiewicz,
- Pseudoprimes and their generalizations (A. Rotkiewicz,
- Reducibility of polynomials (A. Schinzel)
- The number of non-zero coefficients of the greatest
common divisor of two polynomials with given numbers of non-zero
coefficients (A. Schinzel)
- The Mahler measure and other measures of polynomials
- Weak automorphs of binary forms (A. Schinzel)
- Number of solutions of a linear homogeneous congruence
in a box (A. Schinzel)
- Representation of a multivariate polynomial as a sum of
univariate polynomials in linear forms (A. Schinzel)
- Polynomial and exponential congruences to a prime
modulus (M. Skałba, A. Schinzel)
- Congruences for L-functions (J. Urbanowicz)
- 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)
- J. Browkin (with J. Brzeziński), On sequences
of squares with constant second differences, Canadian Math.
Bulletin 48 (2006), 481--491.
- J. Browkin, Elements of small order in K2F,
II, Chin. Ann. Math. Ser. B 28 (2007), 507--520.
- A. Schinzel, Self-inversive polynomials with all
zeros on the unit circle, Ramanujan Journal 9 (2005), 19--23.
- A. Schinzel (with T. Bolis), Identities which
imply that a ring is Boolean, Bull. Greek Math. Soc. 48 (2003),
- 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.
- A. Schinzel (with I. Aliev and W. M. Schmidt),
On vectors whose span contains a given linear subspace, Monatsh.
Math. 144 (2005), 177-191.
- A. Schinzel, On weak automorphs of binary forms
over an arbitrary field, Dissert. Math. 434 (2005), 48 pp.
- A. Schinzel, On the reduced length of a
polynomial, Functiones et Approximatio 35 (2006), 271--306.
- A. Schinzel (with M. Zakarczemny), On a linear
homogeneous congruence, Colloq. Math. 106 (2006), 283--292.
- A. Schinzel, Reducibility of symmetric
polynomials, Bull. Polish Acad. Sci. Mathematics 53 (2005),
- A. Schinzel (with L. Losonczi), Self-inverse
polynomials of odd degree, Ramanujan Journal 14 (2007), 305--320.
- A. Schinzel, On the reduced length of a
polynomial with real coefficients, II, Functiones et Approximatio
37 (2007), 445--459.
- A. Schinzel, Reducibility of a special
symmetric form, Acta Math. Universitatis Ostraviensis 14 (2006),
- A. Schinzel (with A. Białynicki-Birula),
Representations of multivariate polynomials by sums of univariate
polynomials in linear forms, Colloq. Math. 112 (2008), 201--233.
- M. Skałba, On sets which contain a q-th power residue for almost all prime modules, Colloq. Math.
102 (2005), 67--71.
- M. Skałba, Primes dividing both 2n and
3n-2 are rare, Arch. Math. 84 (2005), 485--495.
- 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.