Series that vanish for infintely many values

Question

How to prove that if i got a series $f \in \mathbb{R}[X,Y]$ that vanishes for all $X,Y \in [-1,1]$ then all his coefficients must be $0$ ?

Thank you for your answers.

Answer 1

See Zeros of polynomial over an infinite field. This answers the question for polynomials. If you speak about power series, converging for all $(x,y)\in [-1,1] \times [-1,1]$, the same holds true by the identity theorem for analytic functions in several variables.

Answer 2

If $f=0$ in $(0,1)^2,$ then all partial derivatives of $f$ at $(0,0)$ are $0.$ Couple that with the following: If $f(x,y) = \sum_{m=0}^{M}\sum_{n=0}^{N} a_{mn}x^my^n,$ then $D_x^mD_y^n f(0,0) = m!n!a_{mn}.$