Force tail reduction in polynomial quotient ring

[rachelplayer]

I'd like to "remove squares" in some polynomials living in a polynomial ring over QQ, in 2 variables: x,y. I tried to implement this by modding out by the ideal (x^2 - x, y^2 - y). Depending on the ordering, the result of .mod() does not always output the polynomial I am looking for.

Without specifying an ordering, everything seems fine:

sage: R1.<x,y> = PolynomialRing(QQ, 2)
sage: I1 = R1.ideal(["x^2 - x", "y^2 - y"])
sage: R1("x^2 + y").mod(I1)
x + y
sage: R1("x + y^2").mod(I1)
x + y

However, when specifying the order lex the reduction of x + y^2 is not as expected:

sage: R2.<x,y> = PolynomialRing(QQ, 2, order="lex")
sage: I2 = R2.ideal(["x^2 - x", "y^2 - y"])
sage: R2("x^2 + y").mod(I2)
x + y
sage: R2("x + y^2").mod(I2)
x + y^2

This issue was reported in sage-support where it was pointed out that it is likely a bug in Singular, or in the Singular interface to Sage.

In particular, using the order lex works when implementation="generic" is also specified:

sage: R3.<x,y> = PolynomialRing(QQ, 2, order="lex", implementation="generic")
sage: I3 = R3.ideal(["x^2 - x", "y^2 - y"])
sage: R3("x^2 + y").mod(I3)
x + y
sage: R3("x + y^2").mod(I3)
x + y

For reference, I am using Sage version 8.6 on macOS Mojave 10.14.3.

PS. see also https://groups.google.com/d/msg/sage-devel/K49-V3BbWbg/pxuoehPvAAAJ

CC: @simon-king-jena @malb @mwageringel

Component: commutative algebra

Keywords: multivariate polynomial, quotient ring, singular

Author: Dima Pasechnik

Branch: 21e4f9a

Reviewer: Markus Wageringel

Issue created by migration from https://trac.sagemath.org/ticket/27508

[dimpase]

comment:1

Thanks, one would put "author" in only if intended to provide a patch soon.

[rachelplayer]

Changed author from Rachel Player to none

[rachelplayer]

comment:3

Thanks, in that case I have removed my name from "author".

[dimpase]

comment:4

Singular understand option "lp" for the lexicographic order.
See https://www.singular.uni-kl.de/Manual/4-0-3/sing_31.htm#SEC43

And if I define R2.<x,y> = PolynomialRing(QQ, 2, order="lp")
then it all works as expect. So the bug is in translating the Sage's option "lex" into Singular's one, should be easy to fix.

[dimpase]

comment:5

The following appears to fix this, although I don't really now why...

--- a/src/sage/libs/singular/ring.pyx
+++ b/src/sage/libs/singular/ring.pyx
@@ -50,7 +50,7 @@ from collections import defaultdict
 order_dict = {
     "dp": ringorder_dp,
     "Dp": ringorder_Dp,
-    "lp": ringorder_lp,
+    "lex": ringorder_lp,
     "rp": ringorder_rp,
     "ds": ringorder_ds,
     "Ds": ringorder_Ds,

"lp" order still works, no idea why.

[kwankyu]

comment:7

The file "src/sage/rings/polynomial/term_order.py" might be related.

[dimpase]

comment:8

This bug is already in Sage 8.1, that's as far back as I went.
Perhaps it was always there, I don't know.

Broken lex order should be a blocker, I think...