What is the difference between "Polynomial" and "Multinomial" in two or more variables? What is the difference between &quot;Polynomial&quot; and &quot;Multinomial&quot; in two or more variables?
Since, by definition:

Multinomial:
An algebraic expression having two or more (unlike) terms is called a Multinomial.

For example:
$5x^2 - 2x$ is a multinomial having $2$ terms,
$5x^3- 2xy + 7y^2$ is a multinomial having $3$ terms,
$7xy - 9yz + 6zx - 7$ is a multinomial having $4$ terms.

Polynomials in two or more variables:
An algebraic expression in two or more variables is called a Polynomial if the Power of every variable in each term is a whole number.

Some books say &quot;Multinomial&quot; is one of the types of &quot;Polynomial&quot;, and the other discuss it in particular.
Is the function $f$ cross &quot;Polynomial&quot; or &quot;Multinomial&quot;? Why?
$$f(x, y)=x y + y^2 + 2 x y^2 + y^3 - 3 x y^3 + x y^4,$$
Under many definitions, polynomials and multinomials are synonyms (as commented above). Under any definition, your $f(x,y)$ is both a polynomial and a multinomial. A monomial is not a polynomial nor a multinomial. Poly $\to$ many, multi $\to$ multiple; neither multiple nor many can refer to one.
Polynomial is used more commonly though my experience is that multinomial is more commonly used when referring to plotted distributions. Whereas in describing a function polynomial is used almost exclusively.
A multinomial is not a type of polynomial.
Polynomials are limited to positive integer powers of variables.
A multinomial can contain, say, square roots of a variable.  Or other irrational functions of variables.