Wyniki wyszukiwania - BazTech

1

Polynomial mappings with small degree

Jelonek Z.

Demonstratio Mathematica

|

2015

|

Vol. 48, nr 2

242--249

EN

Let Xn be an affine variety of dimension n and Yn be a quasi-projective variety of the same dimension. We prove that for a quasi-finite polynomial mapping f : Xn → Yn ,every non-empty component of the set Yn\f(Xn) is closed and it has dimension greater or equal to (…), where (…) is a geometric degree of f. Moreover, we prove that generally, if (…) is any polynomial mapping, then either every non-empty component of the set (…) is of dimension (…) or f contracts a subvariety of dimension (…).

2

Łojasiewicz Exponent of Overdetermined Mappings

Spodzieja S., Szlachcińska A.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2013

|

Vol. 61, no 1

27--34

EN

A mapping F:Rn→Rm is called overdetermined if m>n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping F:Rn→Rm can be reduced to the case m=n.

3

On the key exchange and multivariate encryption with nonlinear polynomial maps of stable degree

Ustimenko V., Wroblewska A.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

|

2013

|

Vol. 13, no. 1

63--80

EN

We say that the sequence gn, n≥3, n→∞ of polynomial transformation bijective mapsof free module Kgn over commutative ring K is a sequence of stable degree if the order of gn is growing with n and the degree of each nonidentical polynomial map of kind gkn is an independent constant c. Transformation b = τgnkτ−1, where τ is the affine bijection, n is large and k is relatively small, can be used as a base of group theoretical Diffie-Hellman key exchange algorithm for the Cremona group C(Kn) of all regular automorphisms of Kn. The specific feature of this method is that the order of the base may be unknown for the adversary because of the complexity of its computation. The exchange can be implemented by tools of Computer Algebra (symbolic computations). The adversary can not use the degree of right handside in bx = d to evaluate unknown x in this form for the discrete logarithm problem. In the paper we introduce the explicit constructions of sequences of elements of stable degree for the cases c = 3 and c = n+2/4 for each commutative ring K containing at least 3 regular elements and discuss the implementation of related key exchange and multivariate map algorithms.

4

The infinity phenomenon for polynomial mapping

Biernat G.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2004

|

Vol. 3, nr 1

9--12

EN

In the present paper I give the formula for the residue at infinity of polynomial mapping with n-variables in the case when the numerator degree in the formula defining residuum is not smaller than the sum of degrees of mapping components - n.

5

The jacobians of lower degrees

Biernat G.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2003

|

Vol. 2, nr 1

19--24

EN

In the present paper we give some relation of the number of zeros of a polynomial mapping in C2 with a jacobian of non-maximal degree and the number of branches at infinity of one coordinate of this mapping.

6

Theoreme des residus dans C2

Biernat G.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2002

|

Vol. 1, nr 1

19--24

FR

Dans cet article on propose la demonstration alternative du theoreme des residus dans C2.

7

The residue at infinity and Bezout's theorem

Biernat G.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2002

|

Vol. 1, nr 1

25--27

EN

In this paper we give an alternative proof, based on properties of the residue at infinity, of Bezout’s theorem in C2.

8

On some characterization of proper polynomial mappings

Rodak T., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2000

|

Vol. 48, no 2

157-164

EN

It is well known that a proper, in the classical topology, polynomial mapping is closed in the Zariski topology. In the paper we prove that the inverse is true. Namely, any non-constant polynomial mapping from [C^n] into [C^m] which is closed in the Zariski topology is proper in the classical topology.

9

A number of points in the set [C^2 difference of sets F(C^2)]

Jelonek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

|

1999

|

Vol. 47, no 3

257-261

EN

In this paper we estimate a number of points of the set [C^2 difference of sets F(C^2)] for a polynomial quasi-finite mapping F : [C^2 --> C^2].