Singular

In general you should use as field, the smallest field containing
all of the coefficients, and never anything such as the reals or complexes
which could introduce round-off error.


And, given that you wrote the defining relation as y^7-x^3*(x-5)^4=0,
you probably meant it to be an integral extension of F[x]
That means you should have tried
ring r=0,(y,x),dp; instead of ring r=0,(x,y),dp;
with dp (or equivalently wp(7,7)) as the monomial ordering.

Had the ideal been generated by y^5-x^3*(x-5)^4, then
ring r=0, (y,x), wp(7,5)
would have been a better choice.

Doug

gcd computations of polynomial with inexact coefficients are very challenging
and not implemented in Singular.
If you want to compute with exact coefficients, define your ring as follows:
ring r=(0,i),(x,y),dp; minpoly=i2+1;
or, even better, as:
ring r=0,(x,y),dp;

I've had issues with this before, I assumed it was because I asked for something too difficult. This isn't likely the case this time. Here's what I did:
> LIB "normal.lib";
> ring R=complex,(x,y),dp;
> ideal I=y^7-(x^3)*((x-5)^4);
> list J=normal(I);
To which it replied:
// Computing the equidimensional decomposition...
[1]:
_[1]=x7-y7-20*x6+150*x5-500*x4+625*x3

// number of components is 1

pause>
// start computation of component 1
--------------------------------
We compute the normalization in the ring
// characteristic : 0 (complex:3 digits, additional 6 digits)
// 1 parameter : i
// minpoly : (i^2+1)
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
Computing the jacobian ideal...

The universal denominator is x5-15*x4+75*x3-125*x2
The original singular locus is
_[1]=x5-15*x4+75*x3-125*x2
_[2]=y6
pause>

// mindeg, exponent, vdim used in 'locAtZero': 5 30 30

? not implemented
? error occurred in or before primdec.lib::sep line 3709: ` poly h=gcd(f,diff(f,var(i)));`
? expected poly-expression. type 'help poly;'
? leaving primdec.lib::sep
skipping text from `;` error at token `)`
? leaving primdec.lib::zeroRad
? leaving primdec.lib::radical
? leaving normal.lib::normalM
? leaving normal.lib::normal


This can't be because it's too difficult, right? I must be doing something wrong. What is it?