Growth at infinity of a polynomial with a compact zero set
The Real Jacobian Conjecture for polynomials of degree 3
We show that every local polynomial diffeomorphism (f,g) of the real plane such that deg f ≤ 3, deg g ≤ 3 is a global diffeomorphism.
On the approximate roots of polynomials
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.
Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables
For every polynomial F in two complex variables we define the Łojasiewicz exponents 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 in terms of the local invariants of singularities of the pencil of projective curves associated with F.
A polynomial with 2k critical values at infinity
We construct a polynomial f:ℂ² → ℂ of degree 4k+2 with no critical points in ℂ² and with 2k critical values at infinity.
The Łojasiewicz gradient inequality in a neighbourhood of the fibre
Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.
Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility
In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a plane branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two variables and necessary conditions which a complex curve has to satisfy in order to be the discriminant of a complex plane branch.
Note on the Newton number.
Formulae for the singularities at infinity of plane algebraic curves.
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