This is a basic question, but the usual definitions of algebraic sets seem quite complicated for a non-specialist.
I wanted to confirm if the following is correct: if we have two nontrivial polynomials $p_i(x_1,\dots,x_n)$, and if the maximal degree of $p_1$ in the variable $x_1$ is higher than the maximal degree of $p_2$ in the same variable, then $\{(x_1,\dots,x_n)\in\mathbb{R}^n:p_1(x_1,\dots,x_n)=p_2(x_1,\dots,x_n)=0\} $ has dimension at most $n-2$.
If yes, how to see this also from the complicated definitions, say in wikipedia? And if no, what further condition should $p_1$ and $p_2$ satisfy for this to be true?
Thanks a lot.