/// <summary>
/// Resultant of this polynomial with <c>x^n-1 mod p</c>.
/// </summary>
///
/// <param name="P">P value</param>
///
/// <returns>Returns <c>(rho, res)</c> satisfying <c>res = rho*this + t*(x^n-1) mod p</c> for some integer <c>t</c>.</returns>
public ModularResultant Resultant(int P)
{
// Add a coefficient as the following operations involve polynomials of degree deg(f)+1
int[] fcoeffs = Coeffs.CopyOf(Coeffs.Length + 1);
IntegerPolynomial f = new IntegerPolynomial(fcoeffs);
int N = fcoeffs.Length;
IntegerPolynomial a = new IntegerPolynomial(N);
a.Coeffs[0] = -1;
a.Coeffs[N - 1] = 1;
IntegerPolynomial b = new IntegerPolynomial(f.Coeffs);
IntegerPolynomial v1 = new IntegerPolynomial(N);
IntegerPolynomial v2 = new IntegerPolynomial(N);
v2.Coeffs[0] = 1;
int da = N - 1;
int db = b.Degree();
int ta = da;
int c = 0;
int r = 1;
while (db > 0)
{
c = Invert(b.Coeffs[db], P);
c = (c * a.Coeffs[da]) % P;
a.MultShiftSub(b, c, da - db, P);
v1.MultShiftSub(v2, c, da - db, P);
da = a.Degree();
if (da < db)
{
r *= Pow(b.Coeffs[db], ta - da, P);
r %= P;
if (ta % 2 == 1 && db % 2 == 1)
r = (-r) % P;
IntegerPolynomial temp = a;
a = b;
b = temp;
int tempdeg = da;
da = db;
temp = v1;
v1 = v2;
v2 = temp;
ta = db;
db = tempdeg;
}
}
r *= Pow(b.Coeffs[0], da, P);
r %= P;
c = Invert(b.Coeffs[0], P);
v2.Multiply(c);
v2.Mod(P);
v2.Multiply(r);
v2.Mod(P);
// drop the highest coefficient so #coeffs matches the original input
v2.Coeffs = v2.Coeffs.CopyOf(v2.Coeffs.Length - 1);
return new ModularResultant(new BigIntPolynomial(v2), BigInteger.ValueOf(r), BigInteger.ValueOf(P));
}
