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The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $r(z,bar{z})$ on $mathbb{C}^3$ with diagonal coefficient matrix. This conjecture describes the possible values for the rank of $r(z,bar{z}) |z|^2$ under the hypothesis that $r(z,bar{z})|z|^2=|h(z)|^2$ for some holomorphic polynomial mapping $h$. Our approach is to connect this problem to the degree estimates problem for proper holomorphic monomial mappings from the unit ball in $mathbb{C}^2$ to the unit ball in $mathbb{C}^k$. DAngelo, Kos, and Riehl proved the sharp degree estimates theorem in this setting, and we give a new proof using techniques from commutative algebra. We then complete the proof of the Sum of Squares Conjecture in this case using similar algebraic techniques.
We develop a method for proving sup-norm and Holder estimates for $overline{partial}$ on wide class of finite type pseudoconvex domains in $mathbb{C}^n$. A fundamental obstruction to proving sup-norm estimates is the possibility of singular complex c
Consider a $2$-nondegenerate constant Levi rank $1$ rigid $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in coordinates $(z, zeta, w = u + iv)$: [ u = Fbig(z,zeta,bar{z},bar{zeta}big). ] The Gaussier-Merker model $u=frac{zbar{z}+ frac{1}{2}
For every $epsilon>0$, we give an $exp(tilde{O}(sqrt{n}/epsilon^2))$-time algorithm for the $1$ vs $1-epsilon$ emph{Best Separable State (BSS)} problem of distinguishing, given an $n^2times n^2$ matrix $mathcal{M}$ corresponding to a quantum measurem
We compute the exact norms of the Leray transforms for a family $mathcal{S}_{beta}$ of unbounded hypersurfaces in two complex dimensions. The $mathcal{S}_{beta}$ generalize the Heisenberg group, and provide local projective approximations to any smoo
We generalize Bonahon and Wongs $mathrm{SL}_2(mathbb{C})$-quantum trace map to the setting of $mathrm{SL}_3(mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to every isotopy class of framed oriented links $K$ in a thick