Our subjects of study include the number of Fqn-rational points of Artin-Schreier curves and hypersurfaces, superelliptic curves, and irreducible polynomials. In addition, we determine the number of affine rational points of an Artin-Schreier type hypersurface.
Finite fields
Trace and Norm
For all c∈Fq andα∈Fqn we have that TrFqn/Fq(ca)=cTrFqn/Fq(α). iii) The map α7→TrFqn/Fq is a linear transformation from Fqn to Fq, where both Fqn and Fq are seen as vector spaces over Fq. For allα∈Fqn we have that TrFqn/Fq(αq)=TrFqn/Fq(α) and NFqn/Fq(αq)=NFqn/Fq(α).
Normal Basis
Cyclotomic Polynomials
Let Fq be the finite field of the characteristic p, n a positive integer not divisible by p and d the smallest positive integer such that qd≡1 (modn). ii) the coefficients of Φn(x) belong to the primitive subfield Fp of Fq;. iii) Φn(x) splits into ϕ(n)d distinct monic irreducible polynomials in Fq[x], all of the same degree. The following results provide some classical results about cyclotomous polynomials, which we will use in the proofs of some results in Chapter 4.
Characters sums
Given a division>2ofqn−1and a multiplicative characterχd ofF∗qn with order d, the following is equivalent:. i) there exists a positive integer such that d|(pr+1);. Letψ be the canonical additive character of Fqn and letχ2 be the quadratic character of F∗qn.
Quadratic forms
Our results provide a characterization of the number of affine rational points of this curve in the extension Fqn of Fq. Let Nn(Qg) denote the number of zeros of Qg inFqn and Nn(Cg) the number of affinite rational points of Cg over¡.
Preliminary results
The following theorem, which can be found as an exercise in [41], describes another representation of the polynomial hk. Polynomialshk will be useful in computing the rank of some circular matrices.
Since F(x) is aFq-linearized and the trace is Fq-linear, we must express monomials of the form x·xql in terms of the basisΓ. Then, for each λ∈Fqn, the number of affine rational points in F2qn of the curve. Consequently, this curve is not a maximum or a minimum with respect to the Hasse-Weil limit.
To determine the number of solutions of Tr(x(xqi−x))=Tr(λ), it is necessary to determine the dimension of the symmetric bilinear form associated with this quadratic form, which is the subject of the following theorem . To determine the dimension of the radical of QFit it is sufficient to calculate the dimension of the radical. According to Theorem 2.17, it is sufficient to find the dimension of linear space determined by the roots of .
Since the degree of h(x) is equal to the dimension of the radical, we conclude that. It follows from Lemma 2.19 that the dimension of the radical of the bilinear symmetric form associated with Qi(x) is v=gcd(pa11···pauu,i). It follows that the number of solutions (2.31) is equal to the number of solutions of the system.
Using Lemma 2.20 and Theorem 2.25, we can determine the number of affinite rational points on the curve yq−y=xqi+1−x2−λ, as shown in the following theorem.
The case i = 1
The following proposition determines the rank of Mn,1 and the determinant of one of its reduced matrices. Let M′n,1 denote the main submatrix of Mn,1 constructed from the first row (Mn,1) rows and columns, then M′n,1 is a reduced matrix of Mn,1 and. To prove this, we expand the determinant of Mn−1 by the first row and obtain the recursive relation.
This implies that the sequence {Ln}n≥2 satisfies a recurrence relation (2.35) with corresponding characteristic polynomial given by z2+2z+1=(z+1)2, which has −1 as double root. The number of Nn(C1)affine rational points in F2qn of curve C1 determined by the equation yq−y=xq+1−x2−λis. This theorem allows us to determine when C1 is minimum or maximum with respect to the Hasse-Weil boundary, as we show in the next sequel.
For the other cases, we first show that it suffices to consider the case kui=d. After that, we get a diagonal block matrix consisting of dmatrices of the form Ml,1, kun=ld. We note that every permutation ρ:Zn→Zn defines a natural action on Fnq, given by the following map.
To do this, we show that the product of the permutation matrix Mϕ,Mϕ−1 and. Using Proposition 2.28 and the fact that the matrix Mn,i is a block diagonal matrix with d blocks equal to the matrix Ml,1, we determine the rank of Mn,i and the determinant of the reduced matrix ˜M′n , I of Mn,i. If n=2i, the number Nn(Ci) of affine rational points inF2qn of the curve is determined by the equation yq−y=xqi+1−x2−λ.
Using Theorem 2.33, we can determine the conditions when the curveCi is maximal (or minimal) with respect to the Hasse-Weil bound.
In this chapter we study the number of Fqn-rational points on the affine curve Xd,a,b given by Eq. As a consequence of our results, we calculate the number of elements α in Fqn such that α and Tr(α) are quadratic residues in Fqn. Although these special cases are well studied, a study of the number of rational points on Xd,a,b has not been provided.
Our goal in this chapter is to study the number of rational points on this curve and to provide bounds and explicit formulas for special cases. We use some classical results on quadratic forms over finite fields to provide an expression for the number of rational points on Xd,a,b in terms of Gaussian sums (Proposition 3.7). Using this expression, we apply results on Gaussian sums to obtain bounds on the number of rational points and on the corresponding conditions on d,.
In Section 3.2 we present some remarks, comments and statements of our main results and give an expression for the number of rational points on the curve Xd,a,b in terms of Gaussian sums.
The number of rational points on the curve X d,a,b
As a consequence of our results, we calculate the number of elements α in Fqn such that both α and Tr(α) are quadratic residues (Theorem 3.14). To calculate the value of the sum of the right side of equation (3.3), we will use the fact that Tr(caxTr(x)) defines a quadratic form from Fqn to Fq. The dimension of the radical of the quadratic form Qc is given by the dimension of the radical of the bilinear form Bc(x,y), i.e. dimension of the subspace generated by elements x∈Fqn such that Bc(x,y)=0 for all y∈Fqn.
Therefore, we can determine δfromBc(x,y) by computing the determinant of the reduced matrix associated with 2H. Then direct manipulations on the rows and columns show that 2H reduces to a matrix of the form In summary, we have that the quadratic character of the determinant δ of the reduced matrix of H is given by.
In this case ηD is the restriction of χd to F∗q and we have thatηD is such that η2D=ηD/2.
Bounds and explict formulas for the number of rational points on X d,a,b . 74
These inequalities together with Proposition 3.7 ensure us the result in the cases where D is odd. Our starting point is the case d=2, for which we present a simple expression for the number of affinite rational points given in the next theorem. This result allows us to calculate the number of rational points in specific curves, as in the following examples.
The following definition, which was introduced in the study of diagonal equations [35], will be useful in our results. We now present one of our main results, which provides a formula for the number of rational points in Xd,a,b in the case where B=0 and some appropriate conditions are required. Using this result, we can obtain a simple expression for the number of rational points in the case where, b and satisfy some constraints.
Similar results to Theorem 3.18 can be obtained for the case B̸=0, as we will see in the next section.
Preparation
Lemmata
While most of them apply to general finite fields, we list them for simplicity in the context of binary fields. 30], Theorem 3.35) Let n be a positive integer and let f ∈Fq[x] be an irreducible polynomial of degreem and exponent e. Then the polynomial f(xn) is irreducible over Fq if and only if the following conditions are met:. iii) if 4parts, then 4partm²−1.
The following classical result, due to Swan, concerns the irreducibility of trinomials over the fieldF2. 44], Corollary 5) Let k be positive integers and assume that exactly one of the elements sn,k is odd. The latter, combined with the fact that g̸∈F2[x], implies that ghas one of the following forms.
We emphasize, since it is strange, that the polynomials in the previous lemma are actually pentanomials, i.e. there is no monomial cancellation.
Main results
In particular, since 4 is an even power of 2, Lemma 4.8 implies that it is odd and all irreducible factors of Φp(x) are trinomials in F4. If s=t, Lemma 1.16 means that the factorization of Φp(x) into irreducible polynomials over F2 coincides with that over F4, so the pair (p,2) is also 3-sparse. The latter contradicts Theorem 4.10 because p>7. Therefore, s=2 and then from Lemma 1.16 we conclude that every irreducible factor Φp(x) over F2 has degree 2 and splits into two irreducible trinomials over F4,.
Since the productF1,t(x)H1,t(x) does not have the termx, it follows that 1∉A∩C. The latter implies that the monomial x2 does not appear as the x·xin RHS product of Eq. Therefore, if denotes the number of pentanoms on the RHS of Eq. 4.2) that has terminx2, it follows that is odd. From Lemma 4.7 and Proposition 4.12, it follows that an irreducible factor of Φp(x) overF4 is of the form x3ℓ+ax3+b. Using the criterion described in Lemma 4.3, we conclude that the polynomials tx3k−1+αiwithi∈{1,2} are irreducible over F4 for every k≥1.
Using the criterion described in Lemma 4.3, we conclude that the polynomials x2·5k−1+αix5k−1+1 with i∈{1,2}, are irreducible over F4 for eachk≥1.
The number of rational points of a class of superelliptic curves
Polynomials over binary fields with sparse factors
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