[Math] What’s the degree of a multivariate polynomial in Artin Algebra

abstract-algebraalgebraic-geometrydefinitionirreducible-polynomialspolynomials

According to Degree of a polynomial, it is highest degree among the monomials. Where is this in Artin Algebra?

  1. In Chapter 11.9, an exercise gives a degree to an irreducible complex polynomial of two variables

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  1. In Chapter 11.9, there's a brief discussion of degrees of complex polynomials of two variables in connection with something called Bézout's bound or Bézout's theorem

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  1. In Chapter 11.2, I don't think there's an explicit definition of degree of a polynomial in several variables

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  1. I thought the exercise in Chapter 11.9 refers to $f$'s degree in x because I think that $f$ has positive degree in $x$ is part of the definition of irreducible in Chapter 11.9, but apparently that may not be the case. There's a line:

Let's assume that the polynomial f is irreducible – that it is not a product of two nonconstant polynomials, and also that it has positive degree in the variable x.

I think it is unclear whether or not "it has positive degree in the variable x." is part of the definition of irreducible.

Here is the context:

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Guesses:

  1. We infer it's highest degree among the monomials because we can write the monomials in a polynomial in several variables as $x^i = x_1^{i_1} \dots x_n^{i_n}$

  2. I think I read in Cox, O'Shea and Little something about monomial orders like choosing which of the $i$'s is highest varies by choice of order. Therefore, Artin does not have a definition (at least thus far) for the degree of a multivariate polynomial and thus:

    • $d$ in the Chapter 11.9 exercise refers to $f$'s degree in x and
    • the discussion after the statement of Theorem 11.9.10 is part of Artin's usual method of discussing terms that are not yet defined, much similar to how he defines a subring before a ring and a subfield before a field.

Best Answer

The degree of a multivariable polynomial always refers to the highest degree of its monomials, unless indicated otherwise. This is what Artin always means when he refers to degree with no further qualification. This includes every single usage that you have quoted except for the one that says "degree in the variable $x$".

When Artin says "degree in the variable $x$", that means instead the degree when you pretend all the other variables are coefficients and you just have a polynomial in $x$. In other words, the "degree in $x$" is the largest power of $x$ appearing in the polynomial.

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