Counting solutions to a system of quadratic form equations over finite fields

The problem of counting the number of solutions to equations over finite fields has been a central topic in the study of finite fields. Let $ q $ be a prime power, and denote by $ {\mathbb F}_{q} $ the finite field with $ q $ elements. In this paper, we address the problem of determining the number of solutions in $ {\mathbb F}_{q}^n $ to a system of quadratic form equations with $ n $ unknowns over $ {\mathbb F}_{q} $, a question initially proposed by Carlitz. By employing methods from character sums over finite fields, we derive explicit formulas for the number of solutions to general systems consisting of multiple quadratic forms. Our results provide a complete solution to Carlitz's problem. Separate treatments are provided for the cases of odd and even characteristics.