AngouriMath
Resultant(AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
The resultant of two polynomials with respect to one variable — the condition
on the others under which the two have a common root in it — or
null where that could not be settled.
on the others under which the two have a common root in it — or
Parameter "left"
The first polynomial.
Parameter "right"
The second polynomial.
Parameter "eliminate"
The variable to eliminate between them.
Returns
A polynomial in the remaining variables, vanishing exactly where the two have a
commoneliminate — or where both of their leading
coefficients in it vanish, which is why an eliminant can carry a root the
original pair does not have.null where either argument is
not a polynomial with rational coefficients in at most eight variables of
degree at most 127, where the sum of the two degrees in
eliminate is above 40, or where the elimination ran past its
budget.
common
coefficients in it vanish, which is why an eliminant can carry a root the
original pair does not have.
not a polynomial with rational coefficients in at most eight variables of
degree at most 127, where the sum of the two degrees in
budget.
Remarks
The determinant of the Sylvester matrix, computed as one by fraction-free
elimination. The remainder-sequence formulations are faster and each carries a
sign convention that is easy to get wrong and gives a plausible-looking wrong
answer when it is; taken as a determinant the convention falls out of the
matrix instead of being imposed on it.
elimination. The remainder-sequence formulations are faster and each carries a
sign convention that is easy to get wrong and gives a plausible-looking wrong
answer when it is; taken as a determinant the convention falls out of the
matrix instead of being imposed on it.
Example
// Eliminating y between a circle and a line leaves the condition on x.
Console.WriteLine(MathS.Polynomials.Resultant("x ^ 2 + y ^ 2 - 1", "x + y - 1", "y"));
// 2 * x ^ 2 - 2 * x
// Two polynomials share a root exactly when their resultant vanishes.
Console.WriteLine(MathS.Polynomials.Resultant("x ^ 2 - 1", "x - a", "x"));
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