Cones between the cones of positive semidefinite forms and sums of squares
-
Charu Goel
and
Salma Kuhlmann
Abstract
For n, d ∈ ℕ, the cone 𝓟n+1,2d of positive semidefinite real forms in n + 1 variables of degree 2d contains the subcone Σn+1,2d of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the Hilbert cases (n + 1, 2d) with n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4).
In this paper, we induce a filtration of intermediate cones between Σn+1,2d and 𝓟n+1,2d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties Vk−n ⊊ … ⊊ Vn ⊊ … ⊊ V0 containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n + 1 variables of degree d. By showing that V0, …, Vn (and Vn+1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σn+1,2d. We moreover prove that, in the non-Hilbert cases of (n + 1)-ary quartics for n ≥ 3 and (n + 1)-ary sextics for n ≥ 2, all the remaining cone inclusions are strict.
Funding statement: The first author acknowledges the support through the Oberwolfach Leibniz Fellows program. The second author is grateful for the Oberwolfach Leibniz Graduate Students award and the support provided by the scholarship program of the University of Konstanz under the Landesgraduiertenfürdergesetz and the Studienstiftung des deutschen Volkes. The third author acknowledges the support of the Ausschuss für Forschungsfragen of the University of Konstanz.
Acknowledgements
We wish to thank Mathematisches Forschungsinstitut Oberwolfach for its hospitality. The authors are thankful to Maria Infusino and Victor Vinnikov for useful discussions. We appreciate the insightful suggestions provided by the referee.
Communicated by: D. Plaumann
References
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Articles in the same Issue
- Frontmatter
- Automorphisms and opposition in spherical buildings of classical type
- Revisiting gradient conformal solitons
- Abelian branched covers of rational surfaces, II
- Annulus configurations in handlebody-knot exteriors
- Cones between the cones of positive semidefinite forms and sums of squares
- The geometry of rank drop in a class of face-splitting matrix products: Part I
- The geometry of rank drop in a class of face-splitting matrix products: Part II
- Triality and automorphisms of principal bundles moduli spaces
Articles in the same Issue
- Frontmatter
- Automorphisms and opposition in spherical buildings of classical type
- Revisiting gradient conformal solitons
- Abelian branched covers of rational surfaces, II
- Annulus configurations in handlebody-knot exteriors
- Cones between the cones of positive semidefinite forms and sums of squares
- The geometry of rank drop in a class of face-splitting matrix products: Part I
- The geometry of rank drop in a class of face-splitting matrix products: Part II
- Triality and automorphisms of principal bundles moduli spaces
