On The Partition Regularity of $ax+by = cw^mz^n$

Csikv\'ari, Gyarmati, and S\'ark\"ozy showed that the equation $x+y = z^2$ is not partition regular (PR) over $\mathbb{N}$ and asked if the equation $x+y = wz$ is PR over $\mathbb{N}$. Bergelson and Hindman independently answered this question in the positive. We generalize this result by giving a partial classification of the $a,b,c \in \mathbb{Z}\setminus\{0\}$ and $m,n \in \mathbb{N}$ for which the equation $ax+by = cw^mz^n$ is PR over $\mathbb{Z}\setminus\{0\}$. We show that if $m,n \ge 2$, then $ax+by = cw^mz^n$ is PR over $\mathbb{Z}\setminus\{0\}$ if and only if $a+b = 0$. Next, we show that if $n$ is odd, then the equation $ax+by = cwz^n$ is PR over $\mathbb{Z}\setminus\{0\}$ if and only if one of $\frac{a}{c}, \frac{b}{c},$ or $\frac{a+b}{c}$ is an $n$th power in $\mathbb{Q}$. We come close to a similar characterization of the partition regularity of $ax+by = cwz^n$ over $\mathbb{Z}\setminus\{0\}$ for even $n$, and we examine some equations whose partition regularity remain unknown, such as $16x+17y = wz^8$. In order to show that the equation $ax+by = cwz^n$ is not PR over $\mathbb{Z}\setminus\{0\}$ for certain values of $a,b,c,$ and $n$, we prove a partial generalization of the criteria of Grunwald and Wang for when $\alpha \in \mathbb{Z}$ is an $n$th power modulo every prime $p$. In particular, we show that for any odd $n$ and any $\alpha,\beta,\gamma \in \mathbb{Q}$ that are not $n$th powers, there exist infinitely many primes $p \in \mathbb{N}$ for which none of $\alpha,\beta,$ and $\gamma$ are $n$th powers modulo $p$. Similarly, we show that for any even $n$ and any $\alpha,\beta,\gamma \in \mathbb{Q}$ that are not $\frac{n}{2}$th powers, with one not an $\frac{n}{4}$th power if $4|n$, there exist infinitely many primes $p \in \mathbb{N}$ for which $\alpha,\beta,$ and $\gamma$ are not $n$th powers modulo $p$. Part of the abstract was removed here.