How fast do polynomials grow on semialgebraic sets?

Abstract

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.

When is the Intersection of Two Finitely Generated Subalgebras of a Polynomial Ring Also Finitely Generated?

“…, q k implies that h k is not a polynomial. This, together with observation (a) and [MN14, Assume ω 1 and ω 2 are positive. Then the weighted degree ω can also be described as follows: take the one dimensional family of curves C ξ := {(x, y) : y ω 1 − ξx ω 2 = 0} parametrized by ξ ∈ C. Each of these curves has one place at infinity, i.e.…”

“…sets that are closures of open sets, and it had been asked whether this was indeed the case. In [MN14] this question had been answered in the negative. Our construction in section 4 provides the basis of a particular class of examples in [MN14] consisting of unions of pairs of (non-standard) tentacles.…”

Toric completions and bounded functions on real algebraic varieties

“…Let S ⊆ R d be a semi-algebraic set. In [9], Mondal and Netzer studied the following filtration on the polynomial ring. For n ≥ 0 let…”

“…For us, this question is particularly relevant in the context of positive polynomials and the moment problem, as it concerns possible degree cancellations in sums of positive polynomials, as explained in Section 5. However, a subtle example due to Mondal and Netzer in [9] (see Example 4.4) implies that the L X,m (S) may have infinite dimension. This construction is complemented by our Theorem 5.5, which combines with the results of [13] to show that if S is basic open of dimension at least 2 and admits an S-compatible toric completion but no non-constant bounded function, then the spaces L X,m (S) are finite-dimensional and, consequently, the moment problem for S is not finitely solvable.…”

Algebraicity of normal analytic compactifications of ℂ2 with one irreducible curve at infinity

“…In this subsection we recall from [Mon14] the basic facts of compactifications of affine varieties via degree-like functions. Recall that X = C 2 in our notation; however the results in this subsection remains valid if X is an arbitrary affine variety.…”