Wyniki wyszukiwania

1

Polar quotients and singularities at infinity of polynomials in two complex variables

100%

Płoski A.

Annales Polonici Mathematici

|

2002

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tom 78

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nr 1

49-58

EN

Using the notion of the maximal polar quotient we characterize the critical values at infinity of polynomials in two complex variables. As an application we give a necessary and sufficient condition for a family of affine plane curves to be equisingular at infinity.

2

A bound for the Milnor number of plane curve singularities

100%

Płoski A.

Open Mathematics

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2014

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tom 12

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nr 5

688-693

EN

Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

3

Pinceaux de courbes planes et invariants polaires

64%

Barroso E., Płoski A.

Annales Polonici Mathematici

|

2004

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tom 83

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nr 2

113-128

EN

We study pencils of plane curves $f_t = f - tl^N$, t ∈ ℂ, using the notion of polar invariant of the plane curve f = 0 with respect to a smooth curve l = 0. More precisely we compute the jacobian Newton polygon of the generic fiber $f_t$, t ∈ ℂ. The main result gives the description of pencils which have an irreducible fiber. Furthermore we prove some applications of the local properties of pencils to singularities at infinity of polynomials in two complex variables.

4

On the approximate roots of polynomials

64%

Gwoździewicz J., Płoski A.

Annales Polonici Mathematici

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1994-1995

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tom 60

|

nr 3

199-210

EN

We give a simplified approach to the Abhyankar-Moh theory of approximate roots. Our considerations are based on properties of the intersection multiplicity of local curves.

5

Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables

64%

Gwoździewicz J., Płoski A.

Colloquium Mathematicum

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2005

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tom 103

|

nr 1

47-60

EN

For every polynomial F in two complex variables we define the Łojasiewicz exponents $ℒ_{p,t}(F)$ measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents $ℒ_{p,t}(F)$ in terms of the local invariants of singularities of the pencil of projective curves associated with F.

6

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Multiplicity and the Lojasiewicz exponent

63%

Płoski A.

Banach Center Publications

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1988

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tom 20

|

nr 1

353-364

7

The Łojasiewicz numbers and plane curve singularities

51%

Barroso E., Krasiński T., Płoski A.

Annales Polonici Mathematici

|

2005

|

tom 87

|

nr 1

127-150

EN

For every holomorphic function in two complex variables with an isolated critical point at the origin we consider the Łojasiewicz exponent 𝓛₀(f) defined to be the smallest θ > 0 such that $|grad f(z)| ≥ c|z|^{θ}$ near 0 ∈ ℂ² for some c > 0. We investigate the set of all numbers 𝓛₀(f) where f runs over all holomorphic functions with an isolated critical point at 0 ∈ ℂ².

8

Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of $C^2$

50%

Płoski A.

Annales Polonici Mathematici

|

1990

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tom 51

|

nr 1

275-281

Ograniczanie wyników

5 Annales Polonici Mathematici

1 Banach Center Publications

1 Colloquium Mathematicum

1 Open Mathematics

8 Płoski A.

2 Barroso E.

2 Gwoździewicz J.

1 Krasiński T.

1 2014

2 2005

1 2004

1 2002

1 1995

1 1990

1 1988

Polar quotients and singularities at infinity of polynomials in two complex variables

A bound for the Milnor number of plane curve singularities

Pinceaux de courbes planes et invariants polaires

On the approximate roots of polynomials

Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables

Multiplicity and the Lojasiewicz exponent

The Łojasiewicz numbers and plane curve singularities

Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of $C^2$