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Linear Dependence Of Powers Of Linear Forms

100%
Sładek A.
Annales Mathematicae Silesianae
|
2015
|
tom 29
|
nr 1
131-138
EN
The main goal of the paper is to examine the dimension of the vector space spanned by powers of linear forms. We also find a lower bound for the number of summands in the presentation of zero form as a sum of d-th powers of linear forms.
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Grothendieck and Witt groups in the reduced theory of quadratic forms

100%
Sładek A.
Annales Polonici Mathematici
|
1980
|
tom 38
|
nr 1
13-25
EN
Abstract. Let F be a formally real field. Denote by G(F) and $G_t(F)$ the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group $G(F)/G_t(F)$. This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for $G(F)/G_t(F)$, which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the final part of the paper wo apply the results to determine completely the structure of the reduced Grothendieck group $G(F)/G_t(F)$ for all fields with |g(F)| ≤ 16, where g(F) is the factor group F*lT(F), T (F) being the subgroup of all totally positive elements of F. All the results concerning Grothendieck groups have their counter-parts for Witt groups and we also state and prove the results iu that case.
3
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Kazimierz Szymiczek (1939-2015)

51%
Czogała A., Kula M., Sładek A.
Banach Center Publications
|
2016
|
tom 108
|
nr 1
11-22
4
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Wild Primes of a Higher Degree Self-Equivalence of a Number Field

51%
Czogała A., Rothkegel B., Sładek A.
Annales Mathematicae Silesianae
|
2016
|
tom 30
|
nr 1
17-38
EN
Let ℓ > 2 be a prime number. Let K be a number field containing a unique ℓ-adic prime and assume that its class is an ℓth power in the class group CK. The main theorem of the paper gives a sufficient condition for a finite set of primes of K to be the wild set of some Hilbert self-equivalence of K of degree ℓ.
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