galois.Poly.derivative(k: int = 1) Poly

Computes the $$k$$-th formal derivative $$\frac{d^k}{dx^k} f(x)$$ of the polynomial $$f(x)$$.

Parameters:
k: int = 1

The number of derivatives to compute. 1 corresponds to $$p'(x)$$, 2 corresponds to $$p''(x)$$, etc. The default is 1.

Returns:

The $$k$$-th formal derivative of the polynomial $$f(x)$$.

Notes

For the polynomial

$f(x) = a_d x^d + a_{d-1} x^{d-1} + \dots + a_1 x + a_0$

the first formal derivative is defined as

$f'(x) = (d) \cdot a_{d} x^{d-1} + (d-1) \cdot a_{d-1} x^{d-2} + \dots + (2) \cdot a_{2} x + a_1$

where $$\cdot$$ represents scalar multiplication (repeated addition), not finite field multiplication. The exponent that is “brought down” and multiplied by the coefficient is an integer, not a finite field element. For example, $$3 \cdot a = a + a + a$$.

References

Examples

Compute the derivatives of a polynomial over $$\mathrm{GF}(2)$$.

In [1]: f = galois.Poly.Random(7); f
Out[1]: Poly(x^7 + x^6 + x^4 + x^3, GF(2))

In [2]: f.derivative()
Out[2]: Poly(x^6 + x^2, GF(2))

# p derivatives of a polynomial, where p is the field's characteristic, will always result in 0
In [3]: f.derivative(GF.characteristic)
Out[3]: Poly(0, GF(2))


Compute the derivatives of a polynomial over $$\mathrm{GF}(7)$$.

In [4]: GF = galois.GF(7)

In [5]: f = galois.Poly.Random(11, field=GF); f
Out[5]: Poly(x^11 + x^9 + 5x^8 + 5x^7 + 6x^6 + 6x^5 + x^4 + x^3 + 3x^2 + 3x + 3, GF(7))

In [6]: f.derivative()
Out[6]: Poly(4x^10 + 2x^8 + 5x^7 + x^5 + 2x^4 + 4x^3 + 3x^2 + 6x + 3, GF(7))

In [7]: f.derivative(2)
Out[7]: Poly(5x^9 + 2x^7 + 5x^4 + x^3 + 5x^2 + 6x + 6, GF(7))

In [8]: f.derivative(3)
Out[8]: Poly(3x^8 + 6x^3 + 3x^2 + 3x + 6, GF(7))

# p derivatives of a polynomial, where p is the field's characteristic, will always result in 0
In [9]: f.derivative(GF.characteristic)
Out[9]: Poly(0, GF(7))


Compute the derivatives of a polynomial over $$\mathrm{GF}(3^5)$$.

In [10]: GF = galois.GF(3**5)

In [11]: f = galois.Poly.Random(7, field=GF); f
Out[11]: Poly(66x^7 + 175x^6 + 76x^5 + 199x^4 + 163x^3 + 108x^2 + 108x + 211, GF(3^5))

In [12]: f.derivative()
Out[12]: Poly(66x^6 + 44x^4 + 199x^3 + 216x + 108, GF(3^5))

In [13]: f.derivative(2)
Out[13]: Poly(44x^3 + 216, GF(3^5))

# p derivatives of a polynomial, where p is the field's characteristic, will always result in 0
In [14]: f.derivative(GF.characteristic)
Out[14]: Poly(0, GF(3^5))