Evaluation of weil sums for some polynomials and associated quadratic forms

We study polynomials as quadratic forms over finite fields of odd characteristic. For these polynomials, we establish a connection between a kind of character sums, called Weil sums, and invariants of quadratic forms. A general method is presented to characterize quadratic forms based on coefficients of polynomials, which will be used for some special cases. The evaluation of some Weil sums will also help construct more bent functions or planar functions on finite fields.