Multi-argument Poly Notation

Question

In complexity theory, how do you interpret multi-argument poly notation, such as $poly(a,b)$ or $poly(a,b,c)$?

Does it mean $poly(x)$ for $x=\max(a,b,c)$?

Answer 1

That would be literally $P(a, b)$, for some multivariate polynomial $P(X,Y)$.

Of course, in that case, $P(a,b)\leq |P(a, b)|\leq Q(a, b)$ for some polynomial $Q$, where all coefficients of $Q$ are positive, and then $Q(a, b)\leq Q(\max(a,b), \max(a, b))=R(\max(a,b))$ for some univariate polynomial $R$, where $R(X)=Q(X,X)$.

Conversely, $R(\max(a,b))$ for some polynomial $R$ with nonnegative coefficients is bounded by $R(a+b)$, so this is $P(a,b)$ where $P(X,Y)=R(X+Y)$.