A decomposition lemma in convex integration via classical algebraic geometry

In this paper, we introduce a decomposition lemma that allows error terms to be expressed using fewer rank-one symmetric matrices than $\frac{n(n+1)}{2}$ within the convex integration scheme of constructing flexible $C^{1,\alpha}$ solutions to a system of nonlinear PDEs in dimension $n\geq 2$, which can be viewed as a kind of truncation of the codimension one local isometric embedding equation in Nash-Kuiper Theorem. This leads to flexible solutions with higher H\"older regularity, and consequently, improved very weak solutions to certain induced equations for any $n$, including Monge-Amp\`ere systems and $2$-Hessian systems. The H\"older exponent of the solutions can be taken as any $\alpha <(n^2+1)^{-1}$ for $n=2,4,8,16$, and any $\alpha<(n^2+n-2\rho(\frac{n}{2})-1)^{-1}$ for other $n$, thereby improving the previously known bound $\alpha<(n^2+n+1)^{-1}$ for $n\geq 3$. Here, $\rho(n)$ is the Radon-Hurwitz number, which exhibits an $8$-fold periodicity on $n$ that is related to Bott periodicity. Our arguments involve novel applications of several results from algebraic geometry and topology, including Adams' theorem on maximum linearly independent vector fields on spheres, the intersection of projective varieties, and projective duality. We also use an elliptic method ingeniously that avoids loss of differentiability.