We study the growth of polynomials on semialgebraic sets. For this purpose we associate a graded algebra to the set, and address all kinds of questions about finite generation. We show that for a certain class of sets, the algebra is finitely generated. This implies that the total degree of a polynomial determines its growth on the set, at least modulo bounded polynomials. We however also provide several counterexamples, where there is no connection between total degree and growth. In the plane, we give a complete answer to our questions for certain simple sets, and we provide a systematic construction for examples and counterexamples. Some of our counterexamples are of particular interest for the study of moment problems, since none of the existing methods seems to be able to decide the problem there. We finally also provide new three-dimensional sets, for which the algebra of bounded polynomials is not finitely generated.

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We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the 'multiplicative' property of filtrations on the corresponding completions and introduce a class of projective completions (of arbitrary affine varieties) which generalizes the construction of toric varieties from convex rational polytopes. As an application we recover (and generalize to varieties over algebraically closed fields of arbitrary characteristics) a 'finiteness' property of divisorial valuations over complex affine varieties proved in [dFEI08]. We also find a formula for the pull-back of the 'divisor at infinity' and apply it to compute the matrix of intersection numbers of the curves at infinity on a class of compactifications of certain affine surfaces.

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which completes works of Kushnirenko [Kus76] and Wall [Wal99]. Given a fixed collection of n convex integral polytopes in R n , we also give an explicit characterization of systems of n polynomials supported at these polytopes which have the maximum number (counted with multiplicity) of isolated zeroes on n , or more generally, on the complement of a union of coordinate subspaces of n ; this completes the program (undertaken by many authors including Khovanskii [Kho78], Huber and Sturmfels [HS97], Rojas [Roj99]) of the extension to n of Bernstein's theorem [Ber75] on number of solutions of n polynomials on ( * ) n . Our solutions to these two problems are connected by the computation of the intersection multiplicity at the origin of n hypersurfaces determined by n generic polynomials.

We study two variants of the following question: "Given two finitely generated C-subalgebras R1, R2 of C[x1, . . . , xn], is their intersection also finitely generated?" We show that the smallest value of n for which there is a counterexample is 2 in the general case, and 3 in the case that R1 and R2 are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of C n and to the moment problem on semialgebraic subsets of R n . The counterexample for the general case is a simple modification of a construction of Neena Gupta, whereas the counterexample for the case of integrally closed subalgebras uses the theory of normal analytic compactifications of C 2 via key forms of valuations centered at infinity.2010 Mathematics Subject Classification. 13F20 (primary); 13A18, 16W70, 14M27, 44A60 (secondary).