We study the expected behavior of Betti numbers arrangements zeros random (distributed according to Kostlan distribution) polynomials in R P n $\mathbb {R}\mathrm{P}^n$ . Using a spectral sequence, we prove an asymptotically exact estimate on number connected components complement s $s$ such hypersurfaces also investigate same problem case where are defined by quadratic polynomials. In this cas...

Abstract We provide an effective estimate on the Betti numbers of loop space a compact manifold which admits finite Grauert tube. It implies polynomial in Chen ( arXiv:2101.04368 , 2021) after taking radius tube to infinity.

We provide a proof for an inequality between volume and LBetti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau’s theory of L-Betti numbers of R-simplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for ...

We prove over some local commutative noetherian rings that the sequence of Betti numbers of every finitely generated module is either eventually constant or has termwise exponential growth.

We study mathematical expectations of Betti numbers of configuration spaces of planar linkages, viewing the lengths of the bars of the linkage as random variables. Our main result gives an explicit asymptotic formulae for these mathematical expectations for two distinct probability measures describing the statistics of the length vectors when the number of links tends to infinity. In the proof ...

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: beta{0} is the number of connected components and beta{k} effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously bee...

We generalize work of Lascoux and Józefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2× 2 minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre a...

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters – the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n t...

In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vrećica, and Z̆ival...

In this paper we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first six Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n = 11, there exists precisely 4 examples in which the Betti numbers depend on the grou...