On the Nagata automorphism.
On the irreducibility of fibres of complex polynomial mappings.
The field of Nash functions and factorization of polynomials
The algebraically closed field of Nash functions is introduced. It is shown that this field is an algebraic closure of the field of rational functions in several variables. We give conditions for the irreducibility of polynomials with Nash coefficients, a description of factors of a polynomial over the field of Nash functions and a theorem on continuity of factors.
Some criteria for the injectivity of holomorphic mappings
We prove some criteria for the injectivity of holomorphic mappings.
Errata to "Multiplicity and the Łojasiewicz exponent" (Ann. Polon. Math. 73 (2000), 257-267)
The Łojasiewicz exponent of subanalytic sets
We prove that the infimum of the regular separation exponents of two subanalytic sets at a point is a rational number, and it is also a regular separation exponent of these sets. Moreover, we consider the problem of attainment of this exponent on analytic curves.
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.
Łojasiewicz Exponent of Overdetermined Mappings
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.
Salomon's Theorem for polynomials with several parameters
Sum of squares and the Łojasiewicz exponent at infinity
Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial , where are sums of squares of polynomials of degree at most p, such that f(x) + h(x) > 0 for x...
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