The speed of a class graphs counts the number on vertex set {1,…,n} inside as function n. In this paper, we investigate for many classes that naturally arise in discrete geometry, example intersection segments or disks plane. While upper bounds follow from Warren's theorem (a variant Milnor and Thom), all previously known lower were obtained ad hoc constructions very specific classes. We prove ...

Let f be a Morse function on a closed manifold M , and v be a Riemannian gradient of f satisfying the transversality condition. The classical construction (due to Morse, Smale, Thom, Witten), based on the counting of flow lines joining critical points of the function f associates to these data the Morse complex M * (f, v). In the present paper we introduce a new class of vector fields (f-gradie...

Let f be a polynomial in n variables over some number field and Z subscheme of affine n-space. The notion motivic oscillation index at was initiated by Cluckers [7] Cluckers-Mustaţǎ-Nguyen [12]. In this paper we elaborate on raise several questions. first one is stability under base extension; question linked to deep understanding the density non-archimedean local fields which Igusa's zeta func...

We address the question of whether Thom-Boardman symbol a map germ is an invariant with respect to bi-Lipschitz right equivalence. give counterexample showing that in general answer negative. prove rank contact invariant. Consequently, first and its length are invariants.