galois.Poly.Roots

classmethod galois.Poly.Roots(roots: ArrayLike, multiplicities: Sequence[int] | ndarray | None = None, field: type[Array] | None = None) → Self

Constructs a monic polynomial over $GF (p^{m})$ from its roots.

Parameters:¶

roots: ArrayLike¶

The roots of the desired polynomial.

multiplicities: Sequence[int] | ndarray | None = None¶

The corresponding root multiplicities. The default is None which corresponds to all ones.

field: type[Array] | None = None¶

The Galois field $GF (p^{m})$ the polynomial is over.

None (default): If the roots are an Array, they won’t be modified. If the roots are not explicitly in a Galois field, they are assumed to be from $GF (2)$ and are converted using galois.GF2(roots).
Array subclass: The roots are explicitly converted to this Galois field using field(roots).

Returns:¶

The polynomial $f (x)$ .

Notes

The polynomial $f (x)$ with $k$ roots ${r_{1}, r_{2}, \dots, r_{k}}$ with multiplicities ${m_{1}, m_{2}, \dots, m_{k}}$ is

\begin{aligned} f (x) & = (x - r_{1})^{m_{1}} (x - r_{2})^{m_{2}} \dots (x - r_{k})^{m_{k}} \\ = a_{d} x^{d} + a_{d - 1} x^{d - 1} + \dots + a_{1} x + a_{0} \end{aligned}

with degree $d = \sum_{i = 1}^{k} m_{i}$ .

Examples

Construct a polynomial over $GF (2)$ from a list of its roots.

In [1]: roots = [0, 0, 1]

In [2]: f = galois.Poly.Roots(roots); f
Out[2]: Poly(x^3 + x^2, GF(2))

# Evaluate the polynomial at its roots
In [3]: f(roots)
Out[3]: GF([0, 0, 0], order=2)

Construct a polynomial over $GF (3^{5})$ from a list of its roots with specific multiplicities.

In [4]: GF = galois.GF(3**5)

In [5]: roots = [121, 198, 225]

In [6]: f = galois.Poly.Roots(roots, multiplicities=[1, 2, 1], field=GF); f
Out[6]: Poly(x^4 + 215x^3 + 90x^2 + 183x + 119, GF(3^5))

# Evaluate the polynomial at its roots
In [7]: f(roots)
Out[7]: GF([0, 0, 0], order=3^5)