Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$.
Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } x\in\mathbb{R}\;\tag{A}$$
(SOL) - In practice, since the expression is a polynomial in $x$, I equate all of its coefficients (which are polynomials in $c$) to zero and try to solve this new system for $c$.
First part: Is there a mathematical procedure/technique that corresponds to (SOL)? I do not know much about algebra but wonder if mathematicians have a special name for (SOL)? Or what information about the coefficients (which are polynomials in $c$) may we infer about from our knowledge of $p(x,y)$?
Second Part: Consider the following generalization of the above equation to the system where $c_0=0$ is given but the $c_j$ for $j>0$ are unknowns: \begin{align*}&\sum_{k=0}^{K} p(x,c_k+(-1)^k x)=\sum_{k=0}^{K} p(c_j+(-1)^j x,c_k+(-1)^k x)\\&\text{ for } j=1,\ldots,k \textbf{ and all } x\in\mathbb{R}\tag{B$_K$} \end{align*}
What mathematical tools/theory I have to use to try to answer the following conjecture: If $p$ has degree $q$ then there is no solution for $B_K$ if $K\neq q-1$.
This is from an e-mail I got from an expert addressing the first part of my question, I'm posting it as a community wiki. Then others can attempt to address the second part of the question.
"Dear Sergio: If I understand your problem correctly, then I think it might be easy to determine whether or not the desired c exists.
Suppose we are given p in R[x,y]. Then form the new polynomial of two variables
$$F(x,c) = p(x,x) + p(x,c-x) - p(c-x,x) - p(c-x,c-x).$$
Here, we are temporarily regarding c as a variable. Now write $F(x,c)$ as a polynomial in x whose coefficients lie in $R[c]$:
$$F(x,c) = F_0(c) + F_1(c) x + F_2(c) x^2 + ... + F_n(c) x^n,$$
where $n$ is the maximal power of x that appears in $F(x,c)$ and $F_0(c)$,...,$F_n(c)$ lie in $R[c]$. Then, if you substitute a specific number $c_0$ into $F(x,c)$, you get the polynomial
$F(x,c_0) = F_0(c_0) + F_1(c_0) x + F_2(c_0) x^2 + ... + F_n(c_0) x^n \in R[x].$
This is the zero polynomial if and only if its coefficients vanish, i.e., $F_0(c_0) = F_1(c_0) x = F_2(c_0) x^2 = ... = F_n(c_0)= 0$. Thus $c_0$ is a root of
$$h(c) = gcd(F_0(c),...,F_n(c)).$$
This is easy to compute in Mathematica or Maple. If $h(c)$ has degree 0, then no such c exists, and if $h(c)$ has positive degree, then a c exists, though it may be complex. But now you have the explicit polynomial $h(x)$ and all you need to do is study whether or not it has a real root."