Testing sets for properness of polynomial mappings

Polynomial Mappings with Small Degree

“…By Theorem 5.4 from [4], there is a generically finite polynomial mapping F : k n Ñ k n , such that F 1 p0q " tlu and all other fibers of F are finite. By the Stein factorization theorem ( see [3, Thm.…”

“…Let X, Y be affine varieties. We have the following description of the set of points at which a dominant, generically-finite polynomial map f : X Ñ Y is not finite ( see [4], [6]): Theorem 2.3. Let X, Y be an affine varieties.…”

On a generic symmetry defect hypersurface

“…It is well-known that the set S f of points at which the mapping f is not finite, is either empty or it is a hypersurface (see [5], [6]). We say that the set S f is the set of non-properness of the mapping f. Definition 2.3.…”

Polar varieties and computation of one point in each connected component of a smooth real algebraic set

“…In Subsection 2.5, we will conclude the proof of Theorem 1 by using a result of [16] to relate the properness property and the above normalization result:…”

“…, n}, let mj ∈ C(X ≤r )[T ] be the monic minimal polynomial of Xj in the algebraic field extension C(X ≤r ) → frac(C[X]/P A ). Lemma 3.10 in [16] states that the set of non-properness for the restriction of πr to V (P A ) is the reunion of the zero-sets of the denominators of the coefficients of the polynomials mj. …”