[Math] Explain Multidegree of a polynomial


Definition (The book Ideals, Varieties and Algorithms, Cox et al, pages 59-60 on 2008 edition):

Let $f=\sum_a a_a x^a$ be a nonzero polynomial in $k[x_1,\ldots, x_n]$ and let > be a monomial order.

The multidegree of f is $\quad\quad$ $multideg(f)=max(\alpha\in\mathbb Z^n_{\geq 0} : a_a\ne 0)$.

Example: $\quad\quad\quad\quad\quad\quad\quad$$multideg(f)=(3,0,0)$ for $f=4xy^2z+4z^2-5x^3+7x^2z^2.$

Please explain the definition: maximising each value in the n-tuple $\alpha$ but with respect to what is the multidegree calculated like in the case of $multidegree(f)$ above? With respect to exponents, monomial count, with respect to variable distribution — thank you for any clarification!

Best Answer

The definition of multidegree depends on a monomial ordering chosen. This is a total order on the set of all monomials that has the following aditional property:

  • For all monomials $m,n$ and $p$, $m < n$ implies $p\cdot m < p\cdot n$

In this case, we say that $<$ is an admissible order. A tipical example of a monomial ordering is the lexicographic order, defined as follows: for $\alpha, \beta\in \mathbb Z^n_{\geq 0}$, $\alpha < \beta$ if and only if the leftmost nonzero entry of $\beta - \alpha$ is positive.

Considering this ordering, $(3,0,0)$ is indeed the multidegree of $$f=4xy^2z+4z^2-5x^3+7x^2z^2,$$ but this may not be the case when you consider other orderings.

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