The moment problem on curves with bumps
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Mathematische Zeitschrift
The moment problem on curves with bumps David P. Kimsey1 · Mihai Putinar1,2 Received: 11 October 2019 / Accepted: 14 September 2020 © The Author(s) 2020
Abstract The power moments of a positive measure on the real line or the circle are characterized by the non-negativity of an infinite matrix, Hankel, respectively Toeplitz, attached to the data. Except some fortunate configurations, in higher dimensions there are no non-negativity criteria for the power moments of a measure to be supported by a prescribed closed set. We combine two well studied fortunate situations, specifically a class of curves in two dimensions classified by Scheiderer and Plaumann, and compact, basic semi-algebraic sets, with the aim at enlarging the realm of geometric shapes on which the power moment problem is accessible and solvable by non-negativity certificates. Mathematics Subject Classification 47A57 · 14P99
1 Introduction Throughout the present note R[x1 , . . . , xd ] denotes the ring of polynomials with real coefficients in d indeterminates. We adopt the standard notation
xγ =
d
γ
x j j and |x| :=
x12 + . . . + xd2 ,
j=1
M. Putinar would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program Complex Analysis, Fall 2019, when work on this paper was undertaken. This work was supported by EPSRC Grant number EP/R014604/1. Both authors are thankful to Professor Daniel Plaumann and Professor Claus Scheiderer for pointing out item (c) in Remark 2.3 and also to Professor Markus Schweighofer for carefully reading the paper and making constructive suggestions.
B
David P. Kimsey [email protected] Mihai Putinar [email protected]
1
School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
2
Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106-3080, USA
123
D. P. Kimsey, M. Putinar
where x = (x1 , . . . , xd ) ∈ Rd and γ = (γ1 , . . . , γd ) ∈ Nd0 . The convex cone of polynomials p ∈ R[x1 , . . . , xd ] which can written as a sum of squares is 2 . The elements of 2 represent universally non-negative polynomials. The real zero set of the ideal I := ( p1 , . . . , pk ) generated by p1 , . . . , pk in R[x1 , . . . , xd ] is V (I ) := {x ∈ Rd : p1 (x) = . . . = pk (x) = 0}.
Recalling some basic notions of real algebraic geometry is also in order. Specifically, for a finite subset R = {r1 , . . . , rk } ⊆ R[x1 , . . . , xd ], we let Q R stand for the quadratic module generated by R: Q R = {σ0 + r1 σ1 + . . . + rk σk : σ0 , . . . , σk ∈ 2 }. Also, K Q := {x ∈ Rd : r j (x) ≥ 0 for j = 1, . . . , k} is the common non-negativity set of elements of Q = Q R . In general a quadratic module is a subset of the polynomial algebra closed under addition and multiplication by sums of squares, see [7]. Given a multisequence s = (sγ )γ ∈Nd and a closed set K ⊆ Rd , the full K -moment 0
problem on Rd entails determining whether or not there exists a positive Bor
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