Nullstellens\"atze and Applications

In this expository paper, we present simple proofs of the Classical, Real, Projective and Combinatorial Nullstellens"atze. Several applications are also presented such as a classical theorem of Stickelberger for solutions of polynomial equations in terms of eigenvalues of commuting operators, construction of a principal ideal domain which is not Euclidean, Hilbert’s ${17}^{th}$ problem, the Borsuk-Ulam theorem in topology and solutions of the conjectures of Dyson, Erd"{o}s and Heilbronn.