New Korkin-Zolotarev inequalities

Korkin and Zolotarev showed that if $$\sum_i A_i\Big(x_i-\sum_{j>i} \alpha_{ij}x_j\Big)^2$$ is the Lagrange expansion of a Korkin–Zolotarev (KZ-) reduced positive definite quadratic form, then $A_{i+1}\geq \frac{3}{4} A_i$ and $A_{i+2}\geq \frac{2}{3}A_i$. They showed that the implied bound $A_{5}\geq \frac{4}{9}A_1$ is not attained by any KZ-reduced form. We propose a method to optimize numerically over the set of Lagrange expansions of KZ-reduced quadratic forms using a semidefinite relaxation combined with a branch and bound process. We use a rounding technique to derive exact results from the numerical data. Applying these methods, we prove several new linear inequalities on the $A_i$ of any KZ-reduced form, one of them being $A_{i+4}\geq (\frac{15}{32}-2 \cdot 10^{-5})A_i$. We also give a form with $A_{5}= \frac{15}{32}A_1$. These new inequalities are then used to study the cone of outer coefficients of KZ-reduced forms, to find bounds on Hermite's constant, and to give better estimates on the quality of $k$-block KZ-reduced lattice bases.