# Does “standard polynomial form” define an ordering of terms of equal order e.g. $xy^2 + x^2y$

"Standard polynomial form" defines the ordering of terms to be by their power. For polynomials of one variable, this defines a unique ordering. e.g. $x^2 + x + 1$

With more than one variable, the "order" of a term is the sum of the powers, but this doesn't define a unique ordering. e.g. the terms in $a^3 + a^2b + ab^2 + b^3$ are all third order, so could be written in any ordering.

My question is about the definition of "standard polynomial form": does it define an ordering for the terms of polynomials of more than one variablle?

A secondary question is, if standard form does not actually define an ordering, is there a general ordering by convention?

In the above example, terms are ordered by the power of $a$. This could be generalized by first ordering by the power of the first variable alphabetically, if there are several, then order amongst them by the power of the next variable alphabetically etc. (There's probably a more concise way of describing this idea). e.g. $ab^4c + ab^3c^2 + ab^3c$

I notice that expansion of binomial powers $(a+b)^n$ are written in this order, but that's only one case.

• Take a look at the Mathematica documentation for polynomial ordering. This is the gold-standard computer algebra system. It's not unreasonable to consider the choices Mathematica makes for the default sorting of polynomial terms as trying to adhere to a certain "standard". That said, the system provides a lot of flexibility in changing term-ordering to fit one's particular needs, so "standard" is something of a imprecise descriptor. – Blue Jul 30 '18 at 5:04
• Something to consider: If the polynomial is effectively a random jumble of terms, it's not really going to matter much how you display it; the reader's eyes are just going to glaze over, anyway. If/when the polynomial is trying to tell us all something, find a way to highlight that something. Which of the following would you rather see? $$h^2j^2 k^2-j^2w^2x^2-j^2w^2y^2-k^2x^2y^2+2w^2x^2y^2+2h^2wxyz+2j^2wxyz+2k^2wxyz-2w^3xyz-2wx^3yz-2wxy^3z-k^2w^2z^2-j^2x^2z^2+2w^2x^2z^2-h^2y^2z^2+2w^2y^2z^2+2x^2y^2z^2-2wxyz^3$$ or $$h^2j^2k^2-2(wx-yz) (wy-zx) (wz-xy)-h^2(wx-yz)^2-j^2(wy-zx)^2-k^2(wz-xy)^2$$ – Blue Jul 30 '18 at 5:26

## 1 Answer

There are a few conventions. Some of them are

• Lexicographic order.
• Graded lexicographic order: first compare the degree, then arrange by lexicographic ordering.
• Graded reverse lexicographic order: first compare the degree, then arrange by descending lexicographic ordering.
• Elimination order.
• Weight order.

You can read more about the ordering here.

Also note that "Once a monomial ordering is fixed, the terms of a polynomial (product of a monomial with its nonzero coefficient) are naturally ordered by decreasing monomials (for this order)." (Citation: Gröbner basis).

I am not aware that there is a "standard form" defined.