On the Łojasiewicz exponent of the gradient of a polynomial function

Andrzej Lenarcik

Displaying similar documents to “On the Łojasiewicz exponent of the gradient of a polynomial function”

On the Łojasiewicz exponent of the gradient of a holomorphic function

Andrzej Lenarcik (1998)

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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 ${| g r a d h (x, y) | \geq c | (x, y) |}^{λ}$ holds near $0 \in C^{2}$ 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.

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Agrawal, Kayal, and Saxena recently introduced a new method of proving that an integer is prime. The speed of the Agrawal-Kayal-Saxena method depends on proven lower bounds for the size of the multiplicative semigroup generated by several polynomials modulo another polynomial $h$ . Voloch pointed out an application of the Stothers-Mason ABC theorem in this context: under mild assumptions, distinct polynomials $A, B, C$ of degree at most $1.2 deg h - 0.2 deg rad A B C$ cannot all be congruent modulo $h$ . This paper presents two...

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We discuss the metric theory of simultaneous diophantine approximations in the non-archimedean case. First, we show a Gallagher type 0-1 law. Then by using this theorem, we prove a Duffin-Schaeffer type theorem.