Opened 2 years ago

# Force tail reduction in polynomial quotient ring — at Initial Version

Reported by: Owned by: gh-rachelplayer major sage-9.1 commutative algebra multivariate polynomial, quotient ring, singular SimonKing, malb, gh-mwageringel Rachel Player N/A

### Description

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.

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