# suggestion for documentation of Multivariate Polynomials

It might be helpful for non-experts if the documentation of Multivariate Polynomials were to mention explicitly that the ring being constructed is not the polynomial ring, but the polynomial ring wrt the term ordering, which is not always the same as the polynomial ring.

I found an explanation of the difference in Greuel & Pfister, "A Singular Introduction to Commutative Algebra".

My ignorance of the dependence of the ring on the term ordering was the source of my earlier question:

Daniel Friedan

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Ideally PolynomialRing should be changed to disallow local term orderings because the resulting ring is not a polynomial ring but rather a localization. In the meantime, I agree that a warning is appropriate.

( 2020-07-27 12:19:05 +0100 )edit

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Can you say what documentation page would need a rephrasing?

And where in Greuel & Pfister's book that is explained?

Would you propose an edit to the documentation?

One way to do that online in a few simple steps:

• browse the SageMath documentation on multivariate polynomials
• decide what part of the documentation needs editing
• visit https://gitlab.com/sagemath/sage
• make sure you are on the develop branch (see the menu below the thick blue line)
• navigate to the file that needs editing
• the "Search or jump to..." box at the very top can help find the right file to edit
• make the edits you want to propose
• then send a merge request

A bot then opens a ticket on the Sage Trac server where the proposed change can be reviewed.

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The dependence of the ring on the term ordering is explained in

Gert-Martin Greuel, Gerhard Pfister "A Singular Introduction to Commutative Algebra" with contributions by Olaf Bachmann, Christoph Lossen and Hans Schoenemann Second, Extended Edition Springer-Verlag

Section 1.2 Monomial Orderings p. 16 "SINGULAR Example 1.2.13 (monomial orderings). Global orderings are denoted with a p at the end, referring to “polynomial ring” while local orderings end with an s, referring to “series ring”. "

Section 1.5 Rings Associated to Monomial Orderings p. 38 In this section we show that non–global monomial orderings lead to new rings which are localizations of the polynomial ring.

p. 39 Definition 1.5.1. For any monomial ordering > on Mon(x1,...,xn), we define K[x]> the localization of K[x]

( 2020-07-27 22:01:33 +0100 )edit
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I suggest adding a paragraph in Sage 9.1 Reference Manual: Polynomials, section 1.1 Constructors for Polynomial Rings, immediately after the first paragraph.

But I am singularly unqualified to write the paragraph. I barely grasp the meaning of "localization". I'm only suggesting this revision because I got into trouble because I was ignorant of the fact that "Constructors for polynomial rings" does not mean "Constructors of polynomial rings".

( 2020-07-27 22:07:45 +0100 )edit