Counting rational points of two classes of algebraic varieties over finite fields

Let p stand for an odd prime and let eta is an element of Z(+) (the set of positive integers). Let F-q denote the finite field having q = p(eta) elements and F-q* = Fq \ {0}. In this paper, when the determinants of exponent matrices are coprime to q - 1, we use the Smith normal form of exponent matrices to derive exact formulas for the numbers of rational points on the affine varieties over F-q defined by {a(1)x(1)(d11) ... x(n)(d1n) +... + a(s)x(1)(ds1) ... x(n)(dsn) = b1, a(s+1)x(1)(ds+ 1,1) ...x(n)(ds+1,n) +... + a(s+t)x(1)(ds+ t,1) ... x(n)(ds+ t,n) = b(2) and {c(1)x(1)(e11) ... x(m)(e1m) +... + c(r)x(1)(er1) ...x(m)(erm) = l(1), c(r+1)x(1)(er+1,1) ... x(m)(er+1,m) + ... + c(r+k)x(1)(er+k,1)... x(m)(er+k,m) = l(2), c(r+k+1)x(1)(er+k+1,1) ...x(m)(er+ k+1,m) +... + c(r+k+w)x(1)(er+ k+w,1) ...x(m)(er+ k+w,m) = l(3), respectively, where d(i j), e(i' j') is an element of Z(+), a(i), c(i'). F-q(*), i = 1,..., s + t, j = 1,..., n, i' = 1,..., r + k + w, j' = 1,..., m, and b(1), b(2), l(1), l(2), l(3) is an element of F-q. These formulas extend the theorems obtained by Q. Sun in 1997. Our results also give a partial answer to an open question posed by S.N. Hu, S.F. Hong and W. Zhao [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015), 135-153].