CONSTRUCTING IRREDUCIBLE POLYNOMIALS WITH PRESCRIBED LEVEL CURVES OVER FINITE FIELDS

We use Eisenstein’s irreducibility criterion to prove that there exists an absolutely irreducible polynomial P(X,Y) ∈ GF(q)[X,Y] with coefficients in the finite field GF(q) with q elements, with prescribed level curves Xc := {(x,y)∈GF(q)2 | P(x,y)= c}. 2000 Mathematics Subject Classification. 11T06.


Introduction.
Let GF (q) be the finite field with q elements.Assume that for any c ∈ GF (q), a subset X c (possibly empty) of the finite affine plane GF (q) 2 is given, such that X c ∩ X d ≠ ∅ for any c ≠ d and GF (q) 2 = c∈GF (q) X c . ( In this paper, we use Eisenstein's irreducibility criterion to build absolutely irreducible polynomials such that for any c ∈ GF (q) the level curve {(x, y) ∈ GF (q) 2 | P (x,y) = 0} coincides with X c .Note that P (X,Y ) ∈ GF (q)[X, Y ] is called absolutely irreducible if it is irreducible over the algebraic closure of GF (q).If we define a function f : GF (q) 2 → GF (q) taking a constant value c on the set X c for any c ∈ GF (q), it is easy to see that this is equivalent to the fact that there exists an absolutely irreducible polynomial which interpolates the function f .
It is of course well known that there exists a polynomial that interpolates the function f (see [3,Section 7.5] for a general discussion on this topic).Thus, our result can be viewed as a stronger version of this basic fact, going back to Weber [4].
The basic facts about bivariate polynomial interpolation over finite fields that we will need are summarized in the following theorem.
Theorem 1.1.Any function f : GF (q) 2 → GF (q) can be interpolated by some polynomial in two variables.Moreover, there exists a unique polynomial F(X,Y )∈ GF (q)[X, Y ] of degree less than q in both X and Y that interpolates the function f , that is, satisfying F (a, b) = f (a, b) for any (a, b) ∈ GF (q) 2 .Also, any two interpolating polynomials for f are congruent modulo the ideal of GF (q)[X, Y ] generated by X q − X and Y q − Y .
Our main result is the following theorem.Theorem 1.2.Let f : GF (q) 2 → GF (q) be a function.Then there exists an absolutely irreducible polynomial P (X,Y ) ∈ GF (q)[X, Y ] that interpolates the function f .2. Proof of the main result.Let f : GF (q) 2 → GF (q) be an arbitrary function.By Theorem 1.1, there exists a unique interpolating polynomial H(X, Y ) ∈ GF (q)[X, Y ] for f , of degree at most q − 1 in both X and Y .We order where c 0 (X), c 1 (X),...,c q−1 (X) ∈ GF (q)[X, Y ] are of degree at most q − 1.
Clearly, if we add Y q − Y to H(X, Y ), we still get an interpolating polynomial for f , say K(X, Y ), that is, monic in Y .Thus, it will be perfectly legitimate to start with an interpolating polynomial of the form It is well known (see [3,Corollary 2.11]) that there are irreducible polynomials of any degree over a finite field GF (q).Fix such an irreducible polynomial h(X) ∈ GF (q)[X] of degree 2. Clearly h(X) has two roots in the algebraic closure of GF (q), each of them generating the quadratic extension of GF (q).Let α be a root of h(X) in GF (q), the algebraic closure of GF (q).
Our construction is based on replacing each polynomial coefficient d i (X) of (2.2) with a polynomial of the form where u i (X) ∈ GF (q)[X], such that each e i (X) is divisible by h(X) for i = 0,...,q − 1, while e 0 (X) is not divisible by h(X) 2 .Clearly, the polynomial F(X,Y ) we get by performing these replacements will still be an interpolating polynomial for f , by Theorem 1.1.We will then see that F(X,Y ) follows to be absolutely irreducible.We prove that for some choice of u i (X) ∈ GF (q)[X] in (2.3), e i (X) is divisible by h(X), that is, is solvable.Indeed, from the way we defined h(X), X q − X is relatively prime to h(X).Thus, (2.5) is a linear congruence modulo h(X) in the Euclidean ring GF (q)[X] in which the coefficient X q −X of the unknown u i (X) is relatively prime to the modulus h(X).This being the case, a solution u i (X) of (2.5) exists, and is uniquely determined up to a multiple of h(X).It follows that we can select a solution u i (X) of (2.5) which is a polynomial of degree one.This will completely take care of the cases i = 1,...,q − 1.
For the special case i = 0 we are looking for a solution u 0 (X) of (2.5) satisfying the additional requirement This can be done as follows.If the solution u 0 (X) of the i = 0 case of (2.5) already satisfies (2.6) there is nothing to prove.Otherwise, if u 0 (X) satisfies just replace u 0 (X) with u 0 (X) + h(X).This last polynomial will satisfy both (2.5) and (2.6).The last step in our proof will consist in showing that the polynomial F(X,Y ) constructed above is absolutely irreducible.
The key ingredient of this last step is Eisenstein's irreducibility criterion (see [2, Theorem 6.15]), to the effect that if a polynomial with coefficients in some unique factorization domain R, if we can find some irreducible element p ∈ R which divides γ 0 ,...,γ n−1 , does not divide γ n , while p 2 does not divide γ 0 , then P (X) is an irreducible element of R[X].
We view F(X,Y ) as a (monic) polynomial in Y with coefficients in the unique factorization domain GF Pick up the irreducible Since α is a root of h(X), by the way we constructed F(X,Y ) it follows that p(X) divides the polynomial coefficients e 0 (X), e 1 (X),...,e q−1 (X) ∈ GF (q)[X] and p(X) 2  does not divide the free coefficient e 0 (X).Also, the coefficient of the highest power of Y in (2.4) is 1.Thus, we can apply now Eisenstein's criterion to conclude that F(X,Y ) is an irreducible element of the polynomial ring In other words, the interpolating polynomial F(X,Y ) for f is absolutely irreducible.This concludes the proof of our main theorem.
By our construction, the degrees of the polynomial coefficients e 1 (X),...,e q−1 (X) of F(X,Y ) are at most q + 1, the degree of e 0 (X) is at most q + 2, while F(X,Y ) is monic of degree q in Y .
Theorem 1.2 may be seen as a useful tool in the theory of curves over finite fields, since it allows a fairly elementary and efficient construction of equations of absolutely irreducible plane curves over GF (q) with a given set Z ⊂ GF (q) 2 of GF (q)-rational points (we may, for example, apply our construction to the special case in which the level curves are X 0 = Z, X 1 = GF (q) 2 \Z, and X c = ∅ for any c ∈ GF (q)\{0, 1}).Finally, our interpolation result (with a construction based on a different method, though less direct) still holds true for the case of more than two variables (the proof of this will appear in [1]).