In questo lavoro vengono costruite famiglie di 3-folds algebriche e non singolari di tipo generale tali che l'invariante sia il minimo possibile rispetto al genere geometrico , quando si suppone che il morfismo canonico sia birazionale. Per tali 3-folds vale la relazione lineare inoltre l'immagine del morfismo canonico é una varietà di Castelnuovo di .
A mapping is called overdetermined if m > n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping can be reduced to the case m = n.
It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber such that r is the Łojasiewicz exponent of grad(f) near the fiber . We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula...
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.
We consider some variants of Łojasiewicz inequalities for the class of subsets of Euclidean spaces definable from addition, multiplication and exponentiation : Łojasiewicz-type inequalities, global Łojasiewicz inequalities with or without parameters. The rationality of Łojasiewicz’s exponents for this class is also proved.