Moment-theoretic techniques in the analysis of finite algebraic varieties

Professor Raúl Curto; Department of Mathematics, University of Iowa

For a degree 2n real d-dimensional multisequence β ≡ β(2n) = {βi}i∈Zd+,|i|≤2n to have a representing measure μ, it is necessary for the associated moment matrix M(n) to be positive semidefinite, and for the algebraic variety associated to β, Vβ , to satisfy rank M(n) ≤ card Vβ as well as the following consistency condition: if a polynomial p(x) ≡ Σ|i|≤2n aixi vanishes on Vβ , then p(β) := Σ|i|≤2n aiβi = 0. In previous joint work with L. Fialkow and M. Möller², we proved that for the extremal case (rank M(n) = card Vβ ), positivity and consistency are sufficient for the existence of a (unique, rank M(n)-atomic) representing measure.

In subsequent joint work with Seonguk Yoo³ we considered cubic column relations in M(3) of the form (in complex notation) Z3 = itZ +u ¯ Z, where u and t are real numbers. For (u, t) in the interior of a real cone, we proved that the algebraic variety Vβ consists of exactly 7 points, and we then applied the above mentioned solution of the extremal moment problem to obtain a necessary and sufficient condition for the existence of a representing measure. This required a new representation theorem for sextic polynomials in Z and ¯ Z which vanish in the 7-point set Vβ . Our proof of this representation theorem relies on two successive applications of the Fundamental Theorem of Linear Algebra.

More recently, for n ∈ N, with S. Yoo we have considered the algebraic variety V obtained by intersecting n + 1 algebraic curves of degree n in R2, when the leading terms of the associated bivariate polynomials are all different⁴. We have obtained a new proof, based on the Flat Extension Theorem from the theory of truncated moment problems¹, that the cardinality of V cannot exceed (n+1 over 2 ). In some instances, this provides a slightly better estimate than the one given by Bézout’s Theorem. Thus, we have found a momenttheoretic approach to estimate the cardinality of certain algebraic varieties. Our main results on this topic contribute to the growing literature on the interplay between linear algebra, operator theory, and real algebraic
geometry.

References
¹ R.E. Curto and L.A. Fialkow, Solution of the truncated complex moment problem with flat data, Memoirs Amer. Math. Soc. no. 568, Amer. Math. Soc., Providence, 1996.
² R.E. Curto, L.A. Fialkow and H.M. M¨oller, The extremal truncated moment problem, Integral Equations Operator Theory 60(2008), 177–200.
³ R.E. Curto and S. Yoo, Cubic column relations in truncated moment problems, J. Funct. Anal. 266(2014), 1611–1626.
⁴ R.E. Curto and S. Yoo, A moment theoretic estimate for the cardinality of certain algebraic varieties, New York J. Math. 28(2022) 357 – 366.