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.
We prove some criteria for the injectivity of holomorphic mappings.
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.
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.
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.
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...
Page 1