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On the Łojasiewicz exponent of the gradient of a holomorphic function

100%

Lenarcik A.

Banach Center Publications

1998

tom 44

nr 1

149-166

The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality $|grad h(x,y)| ≥ c|(x,y)|^λ$ holds near $0 ∈ C^2$ for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.

On the Łojasiewicz exponent of the gradient of a polynomial function

100%

Lenarcik A.

Annales Polonici Mathematici

1999

tom 71

nr 3

211-239

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Banach Center Publications

2 Lenarcik A.

1 1999

1 1998

Wyniki wyszukiwania

On the Łojasiewicz exponent of the gradient of a holomorphic function

On the Łojasiewicz exponent of the gradient of a polynomial function