Toric Geometry, Toric Codes, (?)ojasiewicz Exponent And Gradient Space With Reducible Sl(2,C) Action

My thesis consists of Four independent parts which base on my published three papers and a submitted paper. All the works are joint work with my coworkers. the first part is about toric codes. Toric codes are algebraic geometry codes, we study the toric codes which come from toric surfaces. In our previous work [Y-Z] which is a addendum of paper [L-S2], we given the complete classification of toric codes of dimension of less than or equal 5. And we find some new properties in that paper. The work In this thesis is a natural continuation of our previous work [Y-Z] and the beautiful paper of Little and Schwarz [L-S2]. We give an almost complete classification of toric surface codes of dimension less than or equal to 6, except a special pair Cp6(4) and Cp6(5) over F8. Also, we find some new properties, for example, we find that there is some relation between some invariant of a toric surface code and its base field. And also we give an example, CP6(5) and CP6(6) over F7, to illustrate that two monomially equivalent toric codes can be constructed from two lattice non-equivalent polygons. Our results were collected in our paper [L-Y-Z] which is submitted to be considered for publication.The second part is about a important invariant of isolated singularities. Let (?)f be a gradient vector field of a weighted homogenous polynomial with isolated critical point at the origin. Let (w1,…,wn) be the weights of f. In this paper, we prove that the Lojasiewicz Exponentθof f is precisely equal to (?) -1. This means that for some constant c,|(?)f(z)|≥c|z|θin a neighborhood of 0, which provides the optimal lower estimate of|(?)f(z)|. This results will appear at Proceeding of AMS(see[T-Y-Z]).The third part deals with a reducible sl(2, C) action on the formal power series ring. The purpose of this work is to confirm a special case of the Yau conjecture: Suppose that sl(2,C) acts on the formal power series ring via (3.1). Then I(f)= (lii) (?) (li2)(?)…(?)(lis) modulo some one dimensional sl(2, C) representations where (li) is an irreducible sl(2, C) representation of li dimension and{li1, li2,…, lis}(?) {li,l2),…,lr).Unlike classical invariant theory which deals only with irreducible action and 1-dimension representations, we treat the reducible action and higher dimensional representations successively. Our results have published at Science in China (see[Y-Y-Z]).