Książki/Rozdziały
http://hdl.handle.net/11089/5356
2020-08-14T08:54:33ZGelfond-Mahler inequality for multipolynomial resultants
http://hdl.handle.net/11089/31354
Gelfond-Mahler inequality for multipolynomial resultants
Gala-Jaskórzynska, Aleksandra; Kurdyka, Krzysztof; Rudnicka, Katarzyna; Spodzieja, Stanisław
Krasiński, Tadeusz; Spodzieja, Stanisław
We give a bound of the height of a multipolynomial resultant in terms of polynomial degrees, the resultant of which applies. Additionally we give a Gelfond-Mahler type bound of the height of homogeneous divisors of a homogeneous polynomial.
2019-01-01T00:00:00ZA family of hyperbolas associated to a triangle
http://hdl.handle.net/11089/31348
A family of hyperbolas associated to a triangle
Zięba, Maciej
Krasiński, Tadeusz; Spodzieja, Stanisław
In this note, we explore an apparently new one parameter family of conics associated to a triangle. Given a triangle we study ellipses whose one axis is parallel to one of sides of the triangle. The centers of these ellipses move along three hyperbolas, one for each side of the triangle. These hyperbolas intersect in four common points, which we identify as centers of incircle and the three excircles of the triangle. Thus they belong to a pencil of conics. We trace centers of all conics in the family and establish a surprising fact that they move along the excircle of the triangle. Even though our research is motivated by a problem in elementary geometry, its solution involves some non-trivial algebra and appeal to effective computational methods of algebraic geometry. Our work is illustrated by an animation in Geogebra and accompanied by a Singular file.
2019-01-01T00:00:00ZRings and fields of constants of cyclic factorizable derivations
http://hdl.handle.net/11089/31347
Rings and fields of constants of cyclic factorizable derivations
Zieliński, Janusz
Krasiński, Tadeusz; Spodzieja, Stanisław
We present a survey of the research on rings of polynomial constants and fields of rational constants of cyclic factorizable derivations in polynomial rings over fields of characteristic zero.
2019-01-01T00:00:00ZA few introductory remarks on line arrangements
http://hdl.handle.net/11089/31346
A few introductory remarks on line arrangements
Szpond, Justyna
Krasiński, Tadeusz; Spodzieja, Stanisław
Points and lines can be regarded as the simplest geometrical objects. Incidence relations between them have been studied since ancient times. Strangely enough our knowledge of this area of mathematics is still far from being complete. In fact a number of interesting and apparently difficult conjectures has been raised just recently. Additionally a number of interesting connections to other branches of mathematics have been established. This is an attempt to record some of these recent developments.
2019-01-01T00:00:00Z