Vandermonde varieties, mirror spaces, and the cohomology of symmetric semi-algebraic sets

摘要: Let $\mathrm{R}$ be a real closed field, $d,k \in \mathbb{Z}_{> 0}$, $\mathbf{y} =(y_1,\ldots,y_d) \mathrm{R}^d$, and let $V_{d,\mathbf{y}}^{(k)}$ denote the Vandermonde variety defined by $p_1^{(k)} = y_1, \ldots, p_d^{(k)} y_d$, where $p_j^{(k)} \sum_{i=1}^{k} X_i^j$. Then, cohomology groups $\mathrm{H}^*(V_{d,\mathbf{y}}^{(k)},\mathbb{Q})$ have structure of $\mathfrak{S}_k$-modules. We prove that for all partitions $\lambda \vdash k$, $3 0, P < \mathcal{P} \subset \mathrm{D}[X_1,\ldots,X_k]^{\mathfrak{S}_k}_{\leq d}$, $\mathrm{D}$ is an ordered domain contained in $\mathrm{R}$, computes isotypic decomposition, as well ranks first $(\ell+1)$ groups, symmetric semi-algebraic set $\Phi$. The complexity this algorithm (measured number arithmetic operations $\mathrm{D}$) bounded polynomial $k$ $\mathrm{card}(\mathcal{P})$ (for fixed $d$ $\ell$). This result contrasts with $\mathbf{PSPACE}$-hardness problem computing just semi algebraically connected components (i.e. rank $0$-th group) general (non-symmetric) case $d \geq 2$, due to Reif.