AI 中文总结
该研究推翻了实根重数给定的多项式胞腔同调集中在至多一个次数的猜想，通过反例关联符号权重枚举式，证明相关结果并提出同调的完整猜想。
AI 中文摘要
我们遵循文献[KSW]，继续研究由给定次数、具有给定实根重数序列的多项式构成的胞腔复形。一项计算机辅助计算推翻了此前的猜想，即这类胞腔闭包的单点紧化的同调集中在至多一个次数上。具体而言，对于ω=(3,1,1,3)、次数d=18的情况，约化同调在次数7处为ℤ²。这一障碍已在符号胞腔计数中显现，其值为-2。我们将该计数与有理符号权重枚举式F_ω(t)=∑_{η≼ω}(-1)^{ℓn(η)}t^{|η|}关联起来。对于该反例，F_ω(t)=-t¹⁴/(1+t²)²，这给出了欧拉示性数的精确线性公式，并迫使总有理贝蒂数无界。我们证明了若干结果，并提出了描述上述同调的完整猜想。
英文摘要
Following \cite {KSW} we continue the study the cellular complexes formed by polynomials of a given degree having a given sequence of multiplicities of real roots. A computer-assisted calculation disproves the earlier conjecture that homology of one point compactification of the closure of such cell is concentrated in at most one degree. Namely, for $ω=(3,1,1,3)$ in degree $d=18$, the reduced homology is $\ZZ^2$ in degree $7$. The obstruction is already visible in a signed cell count, whose value is $-2$. We relate that count to the rational signed weight enumerator $F_ω(t)=\sum_{η\preceqω}(-1)^{\elln(η)}t^{|η|}$. For the counterexample, $F_ω(t)=-t^{14}/(1+t^2)^2$, which gives an exact linear formula for the Euler characteristic and forces the total rational Betti number to be unbounded. We prove a number of results and formulate a complete conjecture describing the above homology.
Comments10 pages, no figures
