> The algorithm in question is described in “Ideals, Varieties, and
> Algorithms” by David Cox, John Little and Donal O’Shea in Section "§3 A
> Division Algorithm in k[x_1 , … , x_n]".
It is not:
sage: A.<x,y> = PolynomialRing(QQ, order="lex")
sage: (x*y^2 + 1).reduce([x*y + 1, y+1])
x + 1
sage: (x*y^2 + 1).reduce([y+1, x*y + 1])
x + 1
whereas Example 1 in §3 gives remainder 2;
sage: A.<x,y> = PolynomialRing(QQ, order="lex")
sage: (x^2*y + x*y^2 + y^2).reduce([x*y - 1, y^2 - 1])
2*x + 1
sage: (x^2*y + x*y^2 + y^2).reduce([y^2 - 1, x*y - 1])
2*x + 1
whereas Example 2 gives remainder x + y + 1.
> My understanding is it does the (naïve?) reduction algorithm:
>
> reducing = True
> while reducing:
> reducing = False
> for expr in I:
> if expr.leading_monomial().divides(cur.leading_monomial()):
> cur -= cur.leading_term() / expr.leading_term() * expr
> reducing = True
It does not:
sage: def travis_red(f, I):
....: cur = f
....: reducing = True
....: while reducing:
....: reducing = False
....: for expr in I:
....: if expr.lm().divides(cur.lm()):
....: cur -= cur.lt() // expr.lt() * expr
....: reducing = True
....: return cur
....:
sage: travis_red(x^2*y + x*y^2 + y^2, [y^2 - 1, x*y - 1])
2*x + y^2
> Singular itself does this (see pp. 51-52 of "A Singular Introduction
> to Commutative Algebra") and the text even speaks of a non-unique "NF
> [normal form] w.r.t. a non-standard basis", illustrating how the
> results are different when the basis is not standard (aka Gröbner)
> and how there is a unique canonical result when the basis is
> standard.
Singular seems to have changed algorithm since. The example 1.6.13
you're referring to computes slightly different:
sage: A.<x,y,z> = PolynomialRing(QQ, order="degrevlex")
sage: f = x^2*y*z + x*y^2*z + y^2*z + z^3 + x*y
sage: f1 = x*y + y^2 - 1
sage: f2 = x*y
sage: f.reduce([f1, f2]) # same as in the example
y^2*z + z^3
sage: f.reduce([f2, f1]) # missing a xz monomial
y^2*z + z^3 - y^2 + 1
Luca
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