The Jacobian Conjecture in case of "non-negative coefficients"

Ludwik M. Drużkowski

Annales Polonici Mathematici, Tome 66 (1997), p. 67-75 / Harvested from The Polish Digital Mathematics Library

Résumé

It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F (x ₁, . . ., x_{n}) = x - H (x) : = (x ₁ - H ₁ (x ₁, . . ., x_{n}), . . ., x_{n} - H_{n} (x ₁, . . ., x_{n}))$ , where $H_{j}$ are homogeneous polynomials of degree 3 with real coefficients (or $H_{j} = 0$ ), j = 1,...,n and H’(x) is a nilpotent matrix for each $x = (x ₁, . . ., x_{n}) \in ℝ^{n}$ . We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case $d e g F^{- 1} \leq {(d e g F)}^{i n d F - 1}$ , where $i n d F : = m a x i n d H^{'} (x) : x \in ℝ^{n}$ . Note that the above inequality is not true when the coefficients of H are arbitrary real numbers; cf. [E3].

Publié le : 1997-01-01

Zbl 0874.13016

EUDML-ID : urn:eudml:doc:269937

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     title = {The Jacobian Conjecture in case of "non-negative coefficients"},
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     volume = {66},
     year = {1997},
     pages = {67-75},
     zbl = {0874.13016},
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Ludwik M. Drużkowski. The Jacobian Conjecture in case of "non-negative coefficients". Annales Polonici Mathematici, Tome 66 (1997) pp. 67-75. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p67bwm/

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