On Thursday, November 17, 2016 at 6:34:46 PM UTC, Han Frederic wrote:
>
> With singular 4 on sage I have now:
>
> sage: P.<a,b,c> = PolynomialRing(QQ,3, order='lex')
> sage: toto=ideal([7*a - 420*c^3 + 158*c^2 + 8*c - 7, 7*b + 210*c^3 -
> 79*c^2 + 3*c, 84*c^4 - 40*c^3 + c^2 + c])
> sage: toto.interreduced_basis?
> sage: g=toto.interreduced_basis()
> sage: g[0].lc()
> 7
> sage: toto.base_ring()
> Rational Field
> sage: g
> [7*a - 420*c^3 + 158*c^2 + 8*c - 7, 7*b + 210*c^3 - 79*c^2 + 3*c, 84*c^4 -
> 40*c^3 + c^2 + c]
>
> while with singular 3 I had:
> [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 -
> 10/21*c^3 + 1/84*c^2 + 1/84*c]
>
> moreover the doc says:
>
> If this ideal is spanned by (f_1, ..., f_n) this method returns
> (g_1, ..., g_s) such that:
>
> * (f_1,...,f_n) = (g_1,...,g_s)
>
> * LT(g_i) != LT(g_j) for all i != j
>
> * LT(g_i) does not divide m for all monomials m of
>
> {g_1,...,g_{i-1},g_{i+1},...,g_s}
>
> * LC(g_i) == 1 for all i if the coefficient ring is a field.
>
>
>
> Am I missing something? (I need to know if I should change my doctest
> output or not in #21884)
>
I don't know whether the last line of the docs (about LC()==1) comes from
old Singular docs, or not
(there is nothing like this in
https://www.singular.uni-kl.de/Manual/4-0-3/sing_318.htm#SEC357)
Running this example directly at the Singular 4 prompt gives
> ring r=0,(a,b,c),lp;
> ideal i= 7*a - 420*c^3 + 158*c^2 + 8*c - 7, 7*b + 210*c^3 - 79*c^2 + 3*c,
84*c^4 - 40*c^3 + c^2 + c;
> interred(i);
_[1]=84c4-40c3+c2+c
_[2]=7b+210c3-79c2+3c
_[3]=7a-420c3+158c2+8c-7
which is consistent with what you're getting.
Thus, I'd say, correct example and the doc, too...
HTH
Dima
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