"Source: Text/Ring/FldFunRat.text";
"Line: 1085";
"Date: Fri Sep 26 12:10:11 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/FldFunRat.text, line: 1085
// Example: H45E10 ()
print "Example: H45E10";
ei := GetEchoInput();
SetEchoInput(true);
S<u> := PowerSeriesRing(Rationals());
f :=  [1+u-7*u^2, 6-3*u+1/2*u^2-u^3, 5-u+u^2];
seq := MinimalVectorSequence(f, 5);
 [&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
seq := MinimalVectorSequence(f, 5:Distortion:=[2,0,1]);
[&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
p:=2;
seq := MinimalVectorSequence(f, 5:Distortion:=[2,0,1], Power:=p);
sums := [&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
sums;
mp:= map<S->S|  x :-> (IsWeaklyZero(x) select 0 
    else  &+([Coefficient(x,i)*(S.1)^(p*i) :  i in Exponents(x)]))
      + (ISA(Type(v),RngIntElt) select O((S.1)^(p*v)) 
  else S!0 where v := AbsolutePrecision(x))>;
sums := [&+([mp(Q[i])*f[i]: i in [1..#f]]) : Q in seq];
sums;
SetEchoInput(ei);
