Wyniki wyszukiwania - Biblioteka Nauki

1

The infinity phenomenon for polynomial mapping

100%

Biernat G.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2004

|

tom 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.

2

A set on which the Łojasiewicz exponent at infinity is attained

100%

Chądzyński J. , Krasiński T.

|

1997

|

tom 67

|

nr 2

191-197

EN

We show that for a polynomial mapping $F = (f₁,..., fₘ): ℂ^n → ℂ^m$ the Łojasiewicz exponent $𝓛_∞(F)$ of F is attained on the set ${z ∈ ℂ^n: f₁(z) ·...· fₘ(z) = 0}$.

3

The residue at infinity and Bezout's theorem

100%

Biernat G.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2002

|

tom 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.

4

The jacobians of lower degrees

100%

Biernat G.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2003

|

tom 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.

5

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

80%

Ustimenko V. , Wroblewska A.

|

tom 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.

6

Resultant and the Łojasiewicz exponent

80%

Chądzyński J. , Krasiński T.

|

1995

|

tom 61

|

nr 1

95-100

EN

An effective formula for the Łojasiewicz exponent of a polynomial mapping of ℂ² into ℂ² at an isolated zero in terms of the resultant of its components is given.

7

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

80%

Jelonek Z.

|

1999

|

tom 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].

8

On the Łojasiewicz exponent of the gradient of a polynomial function

80%

Lenarcik A.

|

1999

|

tom 71

|

nr 3

211-239

EN

Let $h = ∑ h_{αβ} X^αY^β$ be a polynomial with complex coefficients. The Łojasiewicz exponent of the gradient of h at infinity is the least upper bound of the set of all real λ such that $|grad h(x,y)| ≥ c|(x,y)|^λ$ in a neighbourhood of infinity in ℂ², for c > 0. We estimate this quantity in terms of the Newton diagram of h. Equality is obtained in the nondegenerate case.

9

Łojasiewicz Exponent of Overdetermined Mappings

80%

Spodzieja S. , Szlachcińska A.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2013

|

tom 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.

10

Theoreme des residus dans C2

80%

Biernat G.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2002

|

tom Vol. 1, nr 1

19--24

FR

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

11

Polynomial mappings with small degree

70%

Jelonek Z.

Demonstratio Mathematica

|

2015

|

tom 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 (…).

12

On some characterization of proper polynomial mappings

51%

Rodak T. , Spodzieja S.

|

2000

|

tom 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.