Relationship between the coefficients of polynomials over $\mathrm{GF}(2^n)$ and weights of Boolean functions represented by them

Boolean functions in $n$ variables are represented by polynomials over $\mathrm{GF}(2^n)$. The relationship between the coefficients of polynomials and the weights of functions are researched. Some formulas for expressing the dependence of the first and the second bits in the binary code of the function weight on the polynomial coefficients are obtained. For weights of bent functions and for their subfunctions, some expressions are also obtained.