2014
DOI: 10.1090/s0002-9939-2014-12367-7
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An application of Macaulay’s estimate to sums of squares problems in several complex variables

Abstract: Several questions in CR geometry lead naturally to the study of bihomogeneous polynomials r(z,z) on C n ×C n for which r(z,z) z 2d = h(z) 2 for some natural number d and a holomorphic polynomial mapping h = (h 1 , . . . , h K ) from C n to C K . When r has this property for some d, one seeks relationships between d, K, and the signature and rank of the coefficient matrix of r. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in C[z … Show more

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Cited by 8 publications
(8 citation statements)
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“…The collection of 1-forms (ω a α ) on S n defines the second fundamental form of the mapping f , denoted Π f : T 1,0 S n × T 1,0 S n → T 1,0 S N /f * T 1,0 S n , as described in [2]. We recall from there that (15) ω a α = ω a α β θ β , ω a α β = ω a β α .…”
Section: The Second Fundamental Form and The Gauss Equationmentioning
confidence: 99%
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“…The collection of 1-forms (ω a α ) on S n defines the second fundamental form of the mapping f , denoted Π f : T 1,0 S n × T 1,0 S n → T 1,0 S N /f * T 1,0 S n , as described in [2]. We recall from there that (15) ω a α = ω a α β θ β , ω a α β = ω a β α .…”
Section: The Second Fundamental Form and The Gauss Equationmentioning
confidence: 99%
“…In another recent paper [15] by Grundmeier and Halfpap, the SOS Conjecture 1.2 was established in the special case where A(z,z) is itself an SOS, i.e., (11) A…”
Section: Introductionmentioning
confidence: 99%
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“…In [GK15], the first two authors prove the Sum of Squares Conjecture when r has signature (P, 0), and [BG21] shows it holds when n = 3 and the coefficient matrix of r is diagonal. The gaps in possible rank described in (2) occur in the non-negative semi-definite case, and it appears that when the coefficient matrix of r has negative eigenvalues, the inequality (1) always holds.…”
Section: Introductionmentioning
confidence: 99%
“…Algebraic Formulation. In [GK15] and [BG21], the first two authors investigate r(z, z) z 2 by translating the problem into one about homogeneous ideals in R = C[z 1 , . .…”
Section: Introductionmentioning
confidence: 99%