A constructive nullstellensatz for univariate polynomials

In this dissertation, we will take an effective approach to prove the Hilbert's Nullstellensatz in a special case where we have univariate polynomials $fi(z)\prime s$ for $i\in \{1,2,...,m\}$. This approach will explicitly construct polynomials $pi(z)\prime s$ for $i\in \{1,2,...,m\}.$ Moreover, we will get the best result on the bounds for the degrees of polynomials $pi(z)\prime s.$ 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 $pi(z)\prime s$ 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 $pi\prime s$ 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.