We enumerate all isotopy classes of degree three Morse polynomials R3 * R1 with nonsingular principal homogeneous parts, proving that there are exactly 37 of them. We also count all 2258 isotopy classes of strictly Morse polynomials R3 * R1 of degree three with the maximal possible number (eight) of real critical points. A main tool in this classification is a combinatorial computer program that formalizes Morse surgeries, local monodromy and Picard-Lefschetz theory.
After investigating the geodesic triangles and their angle <^> sums in Nil and SL2 R geometries, we consider the analogous problem in Sol space, which is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sum of geodesic triangles, and we prove that it can be larger than, less than, or equal to pi. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments, and as a special case of this, we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work, we will use the projective model of Sol described by E. Molnar in 1997.
Let G be a locally compact group with the left Haar measure mG. A probability measure & micro; on G is said to be strictly aperiodic if the support of & micro; is not contained in a proper closed left cosets of G. Let & micro; and v be two commuting (with respect to convolution) probability measures on a compact group G and let 1 Mn(& micro;, v) := n X n + 1 i=0 & micro;i & lowast;vn-i be the Birkhoff average of the pair (& micro;, v). Among other results, we show that if one of these measures is strictly aperiodic, then w & lowast;-lim n ->infinity Mn(& micro;, v) = mH, where H is the closed subgroup of G generated by supp & micro; boolean OR supp v and mH is the measure on G defined by mH(B) = mH(B boolean AND H) for every Borel subset B of G. 2020 MATH. SUBJ. CLASS. 28A33, 43A10, 43A77, 47A35.
Minkowski mixed volume of n subpolytopes D1, ... , Dn of a polytope P subset of Rn clearly does not exceed the normalized volume n! Vol(P). Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face F & subne; P intersects at least dim(F) + 1 of the polytopes Di. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems. Motivated by relaxing the bound dim(F) + 1 to dim(F), we prove a combinatorial formula for the mixed volume of a broad class of semi-interlaced polytopes. This class includes, in particular, the off-coordinate polytopes used in computing algebraic degrees such as Maximum Likelihood, Euclidean Distance, and Polar degrees via the Kouchnirenko-Bernshtein theory. We also present applications of our results to the Arnold monotonicity problem (Problem 1982-16), which concerns the dependence of Milnor numbers on the Newton polyhedra.
We construct an irreducible rational curve of degree 10 in CP2 which has 12 triple points, and a union of three rational quartics with 19 triple points. This gives counter-examples to a conjecture by Dimca, Harbourne, and Sticlaru. We also prove that there exists an analytic family C-u of curves of degree 10 with 12 triple points which tends, as u -> 0, to the union of the dual Hesse arrangement of lines (9 lines with 12 triple points) with an additional line. We hope that our approach to the proof of the latter fact could be of independent interest.
