Opened 7 years ago

# Faster qqbar operations using resultants — at Initial Version

Reported by: Owned by: gagern major sage-6.6 number fields qqbar resultant exactify minpoly mmezzarobba, mjo N/A

### Description

This is a spin-off from comment:31:ticket:16964.

Many operations on algebraic numbers can become painfully slow. Most of these operations can be expressed in terms of resultants, and surprisingly the corresponding computations are sometimes way faster than what Sage currently does. So much faster that I'm not sure whether to consider this ticket here a request for enhancement, or even a defect.

Take for example the difference between two algebraic numbers r1 and r2, which are defined as follows:

sage: x = polygen(ZZ)
sage: p1 = x^5 + 6*x^4 - 42*x^3 - 142*x^2 + 467*x + 422
sage: p2 = p1((x-1)^2)
sage: r1 = QQbar.polynomial_root(p2, CIF(1, (2.1, 2.2)))
sage: r2 = QQbar.polynomial_root(p2, CIF(1, (2.8, 2.9)))

Computing their exact difference takes like forever:

sage: r4 = r1 - r2
sage: %time r4.exactify()
(still running, after more than half an hour)

On the other hand, computing a polynomial which has the difference as one root can be achieved fairly easily using resultants, and the resulting number is obtained in under one second:

sage: a,b = polygens(QQ, 'a,b')
sage: %time p3 = r1.minpoly()(a + b).resultant(r2.minpoly()(b), b)
CPU times: user 62 ms, sys: 0 ns, total: 62 ms
Wall time: 68 ms
sage: rs = [r for f in p3.factor()
....:       for r in f[0].univariate_polynomial().roots(QQbar, False)
....:       if r._value.overlaps(r1._value - r2._value)]
sage: assert len(rs) == 1
sage: r3, = rs
sage: %time r3.exactify()
CPU times: user 599 ms, sys: 0 ns, total: 599 ms
Wall time: 578 ms

One possible root of p3 is b=r2 and a+b=r1 which means a=r1-r2. So eliminating b we get a (reducible, not minimal) polynomial in a which has that difference as one of its roots. I try to identify that by looking at the roots r of the factors f, checking whether they overlap the numeric interval.

The way I understand the current code, most exact binary operations are implemented by exactifying both operands to number field elements, then constructing the union of both number fields, converting both operands to that and performing the operation in there. But there is no reason why the number field for the result should be able to contain the operands. I guess dropping that is the main reason why direct resultant computations are faster.

I propose that we try to build all binary operations on algebraic numbers on resultants instead of union fields. I furthermore propose that we try to build the equality comparison directly on resultants of two univariate polynomials, without bivariate intermediate steps.

I can think of two possible problems. One is that we might be dealing with a special case in the example above, and that perhaps number field unions are in general cheaper than resultants. Another possible problem I can imagine is that the resultant could factor into several distinct polynomials, some of which might share a root. If that were the case, numeric refinement wouldn't be able to help choosing the right factor. Should we perhaps not factor the resultant polynomial, but instead compute roots for the fully expanded form?

I'll try to come up with a branch which implements this approach.

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