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 Title: Towards a Bezout-type Theory of Affine Varieties Authors: Mondal, Pinaki Advisor: Milman, Pierre Department: Mathematics Keywords: Affine varietiesProjective completionBezout theoremToric varietiesDegree like functionFiltrationSemidegree Issue Date: 21-Apr-2010 Abstract: We study projective completions of affine algebraic varieties (defined over an algebraically closed field K) which are given by filtrations, or equivalently, integer valued degree like functions' on their rings of regular functions. For a polynomial map P := (P_1, ..., P_n): X -> K^n of affine varieties with generically finite fibers, we prove that there are completions of the source such that the intersection of completions of the hypersurfaces {P_j = a_j} for generic (a_1, ..., a_n) in K^n coincides with the respective fiber (in short, the completions do not add points at infinity' for P). Moreover, we show that there are finite type' completions with the latter property, i.e. determined by the maximum of a finite number of semidegrees', by which we mean degree like functions that send products into sums. We characterize the latter type completions as the ones for which ideal I of the hypersurface at infinity' is radical. Moreover, we establish a one-to-one correspondence between the collection of minimal associated primes of I and the unique minimal collection of semidegrees needed to define the corresponding degree like function. We also prove an affine Bezout type' theorem for polynomial maps P with finite fibers that admit semidegrees corresponding to completions that do not add points at infinity for P. For a wide class of semidegrees of a `constructive nature' our Bezout-type bound is explicit and sharp. URI: http://hdl.handle.net/1807/24371 Appears in Collections: DoctoralDepartment of Mathematics - Doctoral theses