# find quotient of two multivariate polynomials (which are divisible)

I have two multivariate polynomials, `num`

and `denom`

such that
`denom`

divides `num`

. But I have not been able to get SageMath
to simplify the quotient `num/denom`

.

The worksheet and the two polynomials are available at

- https://cocalc.com/share/513993d46f58...
- https://cocalc.com/share/6884a274c847...
- https://cocalc.com/share/d8904075cf0e...

Sample code is below.

```
sage: dR.<d1, d2, d3, d4, d12, d13, d14, d23, d24, d34, d123, d124, d134, d234, d1234> = PolynomialRing(ZZ, 15, order='lex')
sage: denom = load('denom')
sage: num = load('num')
sage: denom.divides(num)
True
sage: num.reduce(Ideal([denom]))
0
sage: F = num/denom
sage: F.denominator()/denom
1
sage: F.reduce()
sage: F.denominator() / denom
1
sage: num.number_of_terms()
3197
sage: denom.number_of_terms()
64
sage: num.degree()
24
```

As pointed out in the answers, the data was created with one term ordering, while the workbook uses another term ordering.

However, why does the term ordering affect the truth of 'num is contained in the ideal generated by denom' i.e., the value of num.reduce(Ideal[denom]) ? One of the answers refers to "This 'local' term ordering ``. I don't understand how the term ordering affects the structure of the ideals in a polynomial ring over the integers.

This is an unfortunate consequence of the fact that

`PolynomialRing`

uses Singular's`ring`

in the background, without restricting the term ordering. Singular's`ring`

is an ordinary polynomial ring when given a "global" term ordering, but it is rather a localization when given a "local" term ordering. These local orderings should be disallowed on a`PolynomialRing`

, and the Singular functionality for localizations should be exposed in another way.