In mathematics, an Artin–Schreier curve is a plane curve defined over an algebraically closed field of characteristic
by an equation for some rational function
One of the most important examples of such curves is hyperelliptic curves in characteristic 2, whose Jacobian varieties have been suggested for use in cryptography.
[1] It is common to write these curves in the form for some polynomials
More generally, an Artin-Schreier curve defined over an algebraically closed field of characteristic
is a branched covering of the projective line of degree
Such a cover is necessarily cyclic, that is, the Galois group of the corresponding algebraic function field extension is the cyclic group
The fundamental theorem of Artin–Schreier theory implies that such a curve defined over a field
has an affine model for some rational function
we define the rational function
over an algebraically closed field
has partial fraction decomposition for some finite set
, and (possibly constant) polynomial
can be chosen so that the above polynomials have degrees coprime to
If that is the case, we define Then the set
is precisely the set of branch points of the covering
is a polynomial, is ramified at a single point over the projective line.
Since the degree of the cover is a prime number, over each branching point
lies a single ramification point
with corresponding different exponent (not to confused with the ramification index) equal to Since
Therefore, the Riemann–Roch theorem may be used to compute that the genus of an Artin–Schreier curve is given by For example, for a hyperelliptic curve defined over a field of characteristic
decomposed as above, Artin–Schreier curves are a particular case of plane curves defined over an algebraically closed field
by an equation for some separable polynomial
yields a covering map from the curve
to the projective line
Separability of defining polynomial
ensures separability of the corresponding function field extension
, a change of variables can be found so that
It has been shown [2] that such curves can be built via a sequence of Artin-Schreier extension, that is, there exists a sequence of cyclic coverings of curves each of degree
, starting with the projective line.