相关论文: On a motivic invariant of the arc-analytic equival…

We define invariants of the blow-Nash equivalence of real analytic function germs, in a similar way that the motivic zeta functions of Denef & Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real…

We propose a refinement of the notion of blow-Nash equivalence between Nash function germs, which is an analog in the Nash setting of the blow-analytic equivalence defined by T.-C. Kuo. The new definition is more natural and geometric.…

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincar\'e series as a motivic…

To a given analytic function germ $f:(\mathbb{R}^d,0) \to (\mathbb{R},0)$, we associate zeta functions $Z_{f,+}$, $Z_{f,-} \in \mathbb{Z} [[T]]$, defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta…

Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain 'motivic integral', living in a completion of the Grothendieck ring of algebraic varieties. He used this…

It has been recently proved that the arc-analytic type of a singular Brieskorn polynomial determines its exponents. This last result may be seen as a real analogue of a theorem by E. Yoshinaga and M. Suzuki concerning the topological type…

We make progress towards the classification of simple Nash germs invariant under the involution changing the sign of the first coordinate, with respect to equivariant blow-Nash equivalence, which is an equivariant Nash version of…

We address the following question, raised by T. Fukui. Is the corank an invariant of the blow-analytic equivalence between real analytic function germs? We give a partial positive answer in the particular case of the blow-Nash equivalence.…

Earlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the…

Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,\sigma)$, where $X$ is a variety over $k$ and $\sigma$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of…

It is known that the weights of a complex weighted homogeneous polynomial $f$ with isolated singularity are analytic invariants of $(\mathbb C^d,f^{-1}(0))$. When $d=2,3$ this result holds by assuming merely the topological type instead of…

We elaborate notions of integration over the space of arcs factorized by the natural $C^*$-action and over the space of non-parametrized arcs (branches). There are offered two motivic versions of the zeta function of the classical monodromy…

We offer an equivariant analogue of the monodromy zeta function of a germ invariant with respect to an action of finite group G as an element of the Grothendieck ring of finite (Z x G)-sets. We formulate equivariant analogues of the…

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant,…

We define a generalised Euler characteristic for arc-symmetric sets endowed with a group action. It coincides with equivariant homology for compact nonsingular sets, but is different in general. We lay emphasis on the particular case of…

We address the question of the classification under blow-Nash equivalence of simple Nash function germs. We state that this classification coincides with the real analytic classification. We prove moreover that a simple germ can not be…

To a given real polynomial function f $\in$ R[x1, . . . , x d ], we associate real topological zeta functions Ztop,0(f\,; s) and Z $\pm$ top,0 (f\,; s) $\in$ Q(s), analogous to the topological zeta function of Denef and Loeser in the…

We offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the Grothendieck ring of finite G-sets tensored by the field of rational numbers. Main two ingredients of the…

In arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced…

We prove the existence of a power structure over the Grothendieck ring of geometric dg categories. We show that a conjecture by Galkin and Shinder (proved recently by Bergh, Gorchinskiy, Larsen, and Lunts) relating the motivic and…