# [Math] Explain Multidegree of a polynomial

algebraic-geometrypolynomialsterminology

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: $$\quad\quad\quad\quad\quad\quad\quadmultideg(f)=(3,0,0)$$ 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!

• 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.