In this paper we study the cones corresponding to sums of squares of n-ary d-ic forms with at most k terms. We show that these are strictly nested as k increases, leading to the usual sum of squares cone. We also discuss the duals of these cones. For n ≥ 3, we construct indefinite irreducible n-ary d-ic forms with exactly k terms for 2 ≤ k ≤ n+d-1n-1.
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Goel et al. (2024) studied this question.
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