Unique Representation in Polynomial Forms

Theorem
Let $$(R,I_x,f)$$ be a polynomial, and $$d=\deg(f)$$

Let $$P_i$$ be the set of all  mononomials $$g$$ over  $$R$$ in the indeterminate  $$I_X$$ such that $$g(X^j)=0$$  for all $$j\neq i$$.

Then there exist $$g_i\in P_i$$,  $$i=0,\ldots,d$$ such that their  sum satisfies


 * $f=g_0+g_1+\cdots+g_d$.

In particular, the $$g_i$$ are defined by $$g_i(X^i)=f(X^i)$$ and $$g_i(X^j)=0$$ for $$j\neq i$$. Moreover, if


 * $f=h_0+h_1+\cdots+h_r$

with $$h_i\in P_i$$, then $$r\geq  d$$, $$h_i=g_i$$ for  $$i=0,\ldots,d$$ and $$h_i$$  is the null polynomial for all  $$i>d$$.