Let \(\mathrm {R}\) be a real closed field. We prove that for each fixed \(\ell , d \ge 0\), there exists an algorithm that takes as input a quantifier-free first-order formula \(\Phi \) with atoms \(P=0, P > 0, P < 0 \text { with } P \in \mathcal {P} \subset \mathrm{D}[X_1,\ldots ,X_k]^{\mathfrak {S}_k}_{\le d}\), where \(\mathrm{D}\) is an ordered domain contained in \(\mathrm {R}\), and computes the ranks of the first \(\ell +1\) cohomology groups, of the symmetric semi-algebraic set defined by \(\Phi \). The complexity of this algorithm (measured by the number of arithmetic operations in \(\mathrm{D}\)) is bounded by a polynomial in k and \(\mathrm {card}(\mathcal {P})\) (for fixed d and \(\ell \)). This result contrasts with the \(\mathbf {PSPACE}\)-hardness of the problem of computing just the zeroth Betti number (i.e., the number of semi-algebraically connected components) in the general case for \(d \ge 2\) (taking the ordered domain \(\mathrm{D}\) to be equal to \(\mathbb {Z}\)). The above algorithmic result is built on new representation theoretic results on the cohomology of symmetric semi-algebraic sets. We prove that the Specht modules corresponding to partitions having long lengths cannot occur in the isotypic decompositions of low-dimensional cohomology modules of closed semi-algebraic sets defined by symmetric polynomials having small degrees. This result generalizes prior results obtained by the authors giving restrictions on such partitions in terms of their ranks and is the key technical tool in the design of the algorithm mentioned in the previous paragraph.

设\(\mathrm {R}\)是一个真正的封闭域。我们证明，对于每个固定的\(\ell , d \ge 0\)，存在一种算法，该算法将无量词的一阶公式\(\Phi \)与原子\(P=0, P > 0, P < 0 \text { with } P \in \mathcal {P} \subset \mathrm{D}[X_1,\ldots ,X_k]^{\mathfrak {S}_k}_{\le d}\)其中\（\ mathrm {d} \）是包含在一个有序的域\（\ mathrm {R} \） ，并且计算所述第一行列\（\ ELL 1 \）的对称半的上同调群，由\(\Phi \)定义的代数集。该算法的复杂度（以\(\mathrm{D}\) 中的算术运算次数来衡量)以k和\(\mathrm {card}(\mathcal {P})\) 中的多项式为界（对于固定d和\(\ell \)）。这个结果与\(\mathbf {PSPACE}\) - 在\(d \ge 2\ )（取有序域\(\mathrm{D}\)等于\(\mathbb {Z}\)）。上述算法结果建立在对称半代数集上同调的新表示理论结果之上。我们证明了在由小次对称多项式定义的封闭半代数集的低维上同调模的同型分解中，不会出现长长度分区对应的 Specht 模。该结果概括了作者获得的先前结果，这些结果在等级方面对此类分区进行了限制，并且是上一段中提到的算法设计中的关键技术工具。