8 ZJ.HUANG,J.S.DENG.Y.Y.FENG AND F.L CHEN -Vim =alin =8)rankM. (4) Dg-a+-小 ={- eodaaeiatadhealammfAanteegsamningteieoftiamide+iri,han 4回- We have the olng Lemma31.LtA3,)-(Aew(a)Ag)),aep≥2a+1md里+y∈R,ha rankAn(.)+ 4e 1 Ano(.u)- 之 dtAn红-rfg,-任-+ 8 Z.J. HUANG, J.S. DENG, Y.Y. FENG AND F.L. CHEN where Z is a column vector consisting of the V (m−α)(n−β) coefficients of the vertex cofactors corresponding to the V interior vertices, and M is a (Ei h(m + 1)(n − β) + Ei v(m − α)(n + 1)) × V (m − α)(n − β) matrix. Then σ can be determined as σ = V (m − α)(n − β) − rankM. (14) 3.3 Dimension formula Suppose p>q 1, define Ap,q(x)=(ai,j ) as a (p + 1) × (p − q) matrix, where ai,j =  0, i<j or i>j + q + 1; q+1 l  xq+1−l , j i j + q + 1, l = i − j. Nonzero elements in each column of Ap,q are the expanding terms of binomial (x + 1)q+1, that is, Ap,q(x) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ xq+1 q+1 1  xq xq+1 . . . . . . ... 1 q+1 q  x ··· xq+1 1 ... q+1 1  xq ... . . . 1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , We have the following Lemma 3.1. Let Ap,q(x, y) =  Ap,q(x) Ap,q(y)  , suppose p 2q + 1 and x = y ∈ R, then rank Ap,q(x, y) = p + 1. Proof. Without loss of generality, assume x = 0. When p = 2q + 1, Ap,q(x, y) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ xq+1 yq+1 q+1 1  xq xq+1 q+1 1  yq yq+1 . . . . . . ... . . . . . . ... 1 q+1 q  x ··· xq+1 1 q+1 q  y ··· yq+1 1 ... q+1 1  xq 1 ... q+1 1  yq ... . . . ... . . . 1 1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ is a (p + 1) × (p + 1) square matrix. In order to calculate det Ap,q(x, y), consider f(z) = (x + z)q+1, g(z)=(y + z)q+1 two polynomials, according to the result about the resultant of two polynomials in [1, p.73], we have det Ap,q(x, y) = res(f, g, z)=(x − y) (q+1)2 . Since x = y, det Ap,q(x, y) = 0, namely, rank Ap,q(x, y) = p + 1.