plumial.core.P._P.ax

_P.ax()

Calculate a pair of polynomials a,x such that x*d = a*k.

This method computes the polynomial factorization that satisfies the fundamental relationship x*d = a*k, where d is the d-polynomial and k is the k polynomial. The factorization separates the d-polynomial ratio d/k into numerator (a) and denominator (x) components.

Return type:

Tuple[Expr, Expr]

Returns:

Tuple of (a_polynomial, x_polynomial) where both are SymPy expressions

Mathematical Relationship:

x * d = a * k a = d / f (where f is the GCD factor) x = k / f

Examples

>>> p = P(133)
>>> a, x = p.ax()
>>> # Verify the relationship symbolically
>>> d_poly = p.d()
>>> k_poly = p.k()
>>> assert sy.expand(x * d_poly) == sy.expand(a * k_poly)