Sum of squares conjecture: the monomial case in $$\mathbb {C}^3$$

Constructing group-invariant CR mappings

“…where ω is a primitive 7-th root of unity. The canonical mapping φ (7,2,4) : S 5 → S 33 associated to (7, 2, 4) is given by…”

“…along with the linear embedding. There has been extensive progress studying rigidity and gap phenomena for sphere maps in recent years, and we refer the reader to [4,9,14,15,[24][25][26][27] and the references therein. As the codimension N −n gets larger, there are many inequivalent maps.…”

Constructing Group-Invariant CR Mappings

“…3 ) along with the linear embedding. There has been extensive progress studying rigidity and gap phenomena for sphere maps in recent years, and we refer the reader to [4,9,14,15,[24][25][26][27] and the references therein. As the codimension − gets larger, there are many inequivalent maps.…”

Algebraic properties of Hermitian sums of squares

“…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.…”

“…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 , . .…”