We give an expression for the {\L}ojasiewicz exponent of a wide class of
n-tuples of ideals $(I_1,..., I_n)$ in $\O_n$ using the information given by a
fixed Newton filtration. In order to obtain this expression we consider a
reformulation of {\L}ojasiewicz exponents in terms of Rees mixed
multiplicities. As a consequence, we obtain a wide class of semi-weighted
homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the
{\L}ojasiewicz of its gradient map $\nabla f$ attains the maximum possible
value.Comment: 25 pages. Updated with minor change

A. Lenarcik

A. Parusiński

A. Płoski

A.G. Kouchnirenko

B. Teissier

C. Bivià-Ausina

C. Huneke

Carles Bivià-Ausina

G. Oleksik

J. Damon

J. Milnor

M. Herrmann

M. Hochster

M. Nagata

M.J. Saia

O. Saeki

S. Łojasiewicz

S.S.-T. Yau

Santiago Encinas

T. Fukui

T. Krasiński

V.I. Arnold

English

arXiv

Crossref

Revista Matemática Complutense

Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the Lojasiewicz exponent of a wide class of n-tuples of ideals (I (1),aEuro broken vertical bar,I (n) ) in using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Lojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions f:(a", (n) ,0)->(a",,0) for which the Lojasiewicz of its gradient map a double dagger f attains the maximum possible value.The first author was partially supported by DGICYT Grant MTM2009-08933. The second author was partially supported by DGICYT Grant MTM2009-07291 and CCG08-UAM/ESP-3928.Bivià-Ausina, C.; Encinas, S. (2013). Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations. Revista Matemática Complutense. 26(2):773-798. doi:10.1007/s13163-012-0104-0S77379826

Bivià-Ausina, Carles

Encinas, Santiago

RiuNet

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the Lojasiewicz exponent of a wide class of n-tuples of ideals (I (1),aEuro broken vertical bar,I (n) ) in using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Lojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions f:(a", (n) ,0)->(a",,0) for which the Lojasiewicz of its gradient map a double dagger f attains the maximum possible value.The first author was partially supported by DGICYT Grant MTM2009-08933. The second author was partially supported by DGICYT Grant MTM2009-07291 and CCG08-UAM/ESP-3928.Bivià-Ausina, C.; Encinas, S. (2013). Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations. 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Lojasiewicz exponent of families of ideals, Rees mixed multiplicities
  and Newton filtrations

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

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