On a question of Moshe Roitman and Euler class of stably free module.

Let A be a ring of dimension d containing an infinite field k , T 1 , ... , T r be variables over A and P be a projective A [ T 1 , ... , T r ] -module of rank n. Assume one of the following conditions holds. (1) 2 n ≥ d + 3 and P is extended from A. (2) 2 n ≥ d + 2 , A is an affine F ‾ p -algebra and P is extended from A. (3) 2 n ≥ d + 3 and singular locus of S p e c (A) is a closed set V (J) with ht J ≥ d − n + 2. Assume U m (P f) ≠ ∅ for some monic polynomial f (T r) ∈ A [ T 1 , ... , T r ]. Then U m (P) ≠ ∅ (see 6.1). [ABSTRACT FROM AUTHOR]