Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces

通过格罗滕迪克的儿童画计算三项式曲线与$\mathbf{T}$-项式曲线的正交点个数

Counting positive intersection points of a trinomial and a $\mathbf{T}$-nomial curves via Groethendieck's dessin d'enfant

本文利用格罗滕迪克的儿童画分析实多项式系统的正孤立解个数，将三项式与$t$项式系统的上界从$2^t-2$改进为$3\cdot 2^{t-2}-1$，并刻画了$t=3$时达到精确界五的几何条件。

Polyhedral divisors of affine trinomial hypersurfaces

We find polyhedral divisors corresponding to the torus action of complexity one on affine trinomial hypersurfaces. Explicit computations for particular classes of such hypersurfaces including Pham-Brieskorn surfaces and rational trinomial hypersurfaces are given.

Polynomial curves on trinomial hypersurfaces

We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwartz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwartz-Halphen's Theorem for Pham-Brieskorn surfaces.

Orbits of automorphism group of trinomial hypersurfaces

Trinomial hypersurfaces form a natural class of affine algebraic varieties closely connected with varieties admitting a torus action of complexity one. We investigate orbits of the automorphism group on these hypersurfaces. We prove that each nonrigid trinomial variety has finite number of orbits. We investigate singular orbits and this gives us description of all orbits for some classes of flexible trinomial hypersurfaces. Also we obtain a description of orbits for a class of hypersurfaces having a unique variable with power one in the equation.

On rigidity of trinomial hypersurfaces and factorial trinomial varieties

Trinomial varieties are affine varieties given by some special system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal rational varieties with torus action of complexity one. For an affine variety X we consider the subgroup SAut(X) of the automorphism group generated by all algebraic subgroups isomorphic to the additive group of the ground field. An affine variety X is rigid if SAut(X) is trivial. In opposite an affine variety is flexible if SAut(X) acts transitively on the regular locus. Arzhantsev proved a criterium for a factorial…

On rigidity of factorial trinomial hypersurfaces

An affine algebraic variety $X$ is rigid if the algebra of regular functions ${\mathbb K}[X]$ admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least 2.