In this dissertation, we will take an effective approach to prove the Hilbert's Nullstellensatz in a special case where we have univariate polynomials fi(z)′s for i∈{1,2,...,m}. This approach will explicitly construct polynomials pi(z)′s for i∈{1,2,...,m}. Moreover, we will get the best result on the bounds for the degrees of polynomials pi(z)′s. We then use a similar technique to solve the problems in a matrix case. Previous work motivated by algebraic techniques are from {[}2{]} W.D.Brownawell, {[}5{]} J.Kollar. They made a big improvement on the bounded degree of pi(z)′s in solutions. We are also motivated by works done in analysis from L. Carleson (1962), T. Wolff (1979). These are used to get the best result on the bounds on the degrees of pi′s in the solutions obtained in this dissertation. For the matrix case, we are motivated by {[}11{]} T.T. Trent, X. Zhang. This will enable us to derive the results in the matrix case.