What is the difference between the "degree" of a polynomial and its "total degree"?
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$\begingroup$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:
$$\deg_1(x_1^2x_2x_3^4-3x_2+4x_1x_4^5-x_1x_2^3x^2)=2\;,\;\;\deg_4(x_1^2x_2x_3^4-3x_2+4x_1x_4^5-x_1x_2^3x^2)=5\ldots$$
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