对称半代数集上同调模的同构分解：重数的多项式界
On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities
摘要：
我们考虑对称（在有限对称群的积的作用下）实代数簇和半代数集，以及仿射和射影空间中的对称复簇，由固定常数$d$限制的次数多项式定义。我们证明了如果一个Specht模$\mathbb{S}^λ$在这类集合的上同调模的同构分解中以正重数出现，则划分$λ$的秩由$O(d)$有界.这意味着一个多项式（在周围空间的维数）上的数量限制这样的模块。此外，我们证明了多项式上的多重性的限制，这些确实出现在上述上同调模的同型分解的正多重性。我们给出了我们的方法在证明某些对称半代数集的定义多项式的次数的下界，以及在（不一定是对称的）有界实代数集的投影下的图像的贝蒂数的改进界中的一些应用，在某些情况下改进了Gabrielov，Vorobjov和Zell的先前结果.
英文摘要：
We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant $d$. We prove that if a Specht module, $\mathbb{S}^λ$, appears with positive multiplicity in the isotypic decomposition of the cohomology modules of such sets, then the rank of the partition $λ$ is bounded by $O(d)$. This implies a polynomial (in the dimension of the ambient space) bound on the number of such modules. Furthermore, we prove a polynomial bound on the multiplicities of those that do appear with positive multiplicity in the isotypic decomposition of the above mentioned cohomology modules. We give some applications of our methods in proving lower bounds on the degrees of defining polynomials of certain symmetric semi-algebraic sets, as well as improved bounds on the Betti numbers of the images under projections of (not necessarily symmetric) bounded real algebraic sets, improving in certain situations prior results of Gabrielov, Vorobjov and Zell.
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