On the speed of algebraically defined graph classes

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 general giving an essentially tight bound whose edges are defined using signs given finite list polynomials, assuming these polynomials satisfy some reasonable conditions. This particular implies different graphs, which match bounds. Our result also gives counting containment orders various families geometric objects, including circle angle orders. Some applications presented paper new, whereas others recover results Alon-Scheinerman, Fox, McDiarmid-Müller Shi. For proof our use tools algebraic geometry differential topology.