"Total" degree of a polynomial?

Question

What is the difference between the "degree" of a polynomial and its "total degree"?

Answer 1

The total degree of a polynomial in more than one variable is the maximal of the sums of all the powers of the variables in one single monomial. For example

$$\deg(x_1^2x_2x_3^4-3x_2+4x_1x_4^5-x_1x_2^3x^2)=7$$

You can also define the local or particular degree for some particular variable, say:

$$\deg_1(x_1^2x_2x_3^4-3x_2+4x_1x_4^5-x_1x_2^3x^2)=2\;,\;\;\deg_4(x_1^2x_2x_3^4-3x_2+4x_1x_4^5-x_1x_2^3x^2)=5\ldots$$