gr_minipoly, minipoly

gr_minipoly(plist,vlist,order,poly,v,homo)

:: Computation of the minimal polynomial of a polynomial modulo an ideal

minipoly(plist,vlist,order,poly,v)

:: Computation of the minimal polynomial of a polynomial modulo an ideal

return

polynomial

plist, vlist

list

order

number, list or matrix

poly

polynomial

v

indeterminate

homo

flag

• gr_minipoly() begins by computing a Groebner basis. minipoly() regards an input as a Groebner basis with respect to the variable order vlist and the order type order.
• Let K be a field. If an ideal I in K[X] is zero-dimensional, then, for a polynomial p in K[X], the kernel of a homomorphism from K[v] to K[X]/I which maps f(v) to f(p) mod I is generated by a polynomial. The generator is called the minimal polynomial of p modulo I.
• gr_minipoly() and minipoly() computes the minimal polynomial of a polynomial p and returns it as a polynomial of v.
• The minimal polynomial can be computed as an element of a Groebner basis. But if we are only interested in the minimal polynomial, minipoly() and gr_minipoly() can compute it more efficiently than methods using Groebner basis computation.
• It is recommended to use a degree reverse lex order as a term order for gr_minipoly().

[117] G=tolex(G0,V,0,V)$      
43.818sec + gc : 11.202sec
[118] GSL=tolex_gsl(G0,V,0,V)$
17.123sec + gc : 2.590sec
[119] MP=minipoly(G0,V,0,u0,z)$
4.370sec + gc : 780msec

References

section lex_hensel, lex_tl, tolex, tolex_d, tolex_tl.

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