Specialization and integral closure

Specialization and integral closure

I ′/(x) = I ′/(x) , where x = ∑n i=1 ziai is a generic element for I defined over the polynomial ring R ′ = R[z1, . . . , zn] and I ′ denotes the extension of I to R. This result can be paraphrased by saying that an element is integral over I if it is integral modulo a generic element of the ideal. Other, essentially unrelated, results about lifting integral dependence have been proved by Teiss...

Computing Integral Closure Workshop

Integral Specialization of Families of Rational Functions

Suppose C is an algebraic curve, f is a rational function on C defined over Q, and A is a fractional ideal of Q. If f is not equivalent to a polynomial, then Siegel’s theorem gives a necessary condition for the set C(Q)∩f−1(A) to be infinite: C is of genus 0 and the fiber f−1(∞) consists of two conjugate quadratic real points. We consider a converse. Let P be a parameter space for a smooth fami...

Real Integral Closure and Milnor Fibrations

We give a condition to guarantee the existence of a Milnor fibration for real map germs of corank 1, which include cases that are not L-maps in the sense of Massey. Our approach exploits the structure of a family of functions.

Nilpotents, Integral Closure and Equisingularity conditions