"Source: Text/Basics/RngMPol.text";
"Line: 1452";
"Date: Thu Sep 25 14:47:46 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Basics/RngMPol.text, line: 1452
// Example: H25E7 ()
print "Example: H25E7";
ei := GetEchoInput();
SetEchoInput(true);
Q := RationalField();
PR<x, y, z> := PolynomialRing(Q, 3);
g := 119/44*x^2 - 93759/41440*x*y + 390935/91427*x*z
     + 212/243*x - 3/17*y^2 + 52808/172227*y*z
     - 287/227*y + 537/934*z^2 - 127/422*z;
SymmetricBilinearForm(g);
DiagonalForm(g);
bl := SymmetricBilinearForm(g);
NBL := Matrix(PR, bl);
D, T := OrthogonalizeGram(bl);
NT := Matrix(PR, T);
C := Matrix(PR, [[x,y,z,1]]);
NC := C * NT;
NCT := Transpose(NC);
(NC * NBL * NCT)[1][1] eq DiagonalForm(g);
assert $1;
SetEchoInput(ei);
