On circulant matrices and rational points of Artin Schreier's curves
aa r X i v : . [ m a t h . N T ] F e b ON CIRCULANT MATRICES AND RATIONAL POINTS OF ARTINSCHREIER CURVES
DANIELA OLIVEIRAF. E. BROCHERO MART´INEZ
Abstract.
Let F q be a finite field with q elements, where q is an odd prime power.In this paper we associated circulant matrices and quadratic forms with the Artin-Schreier curve y q − y = x · P ( x ) − λ, where P ( x ) is a F q -linearized polynomial and λ ∈ F q . Our main results provide a characterization of the number of rational pointsin some extension F q r of F q . In particular, in the case when P ( x ) = x q i − x we givena full description of the number of rational points in term of Legendre symbol andquadratic characters. Introduction
Let F q be a finite field with q = p e elements, with p an odd prime and e a positiveinteger. Algebraic curves over finite fields have many applications to several areas suchthat coding theory, cryptography, communications and related areas (see [8, 14]). Forinstance, elliptic curves over finite fields, that is a class of algebraic curves, have beenused to construct cryptography codes, because their points form an appropriate group. Inany case, it is important to determine the number of points of the curve in each extensionof the field F q . For g ( x ) ∈ F q [ x ], a plane curve with equation of the form C : y q − y = g ( x ) , is called an Artin-Schreier curve. These type of curves have been studied extensively inseveral contexts (see ([4, 5, 11, 15]). Besides that, Artin-Schreier curves over finite fieldsin some cases can be closely related to quadratic forms. We have some characterizationsand classification in the literature about these relations (see [1, 2, 12]).The aim of this paper is to study the number of rational points in F q r ( r ≥
1) of C when g ( x ) = x · f ( x ) and f ( x ) is a F q -linearized polynomial. For that we associated tothe curve C with a quadratic form and in a more general case we associated the numberof rational of C with a circulant matrix. Using that relation and determining the rankof that matrix, we given an explicit formula for the number of rational points in anyextension F q r of F q in the cases when gcd( r, q ) = 1. Our main result for that case providean explicit formula for the number of rational points, that is given in Theorem 3.4.Moreover, in the particular case when the curve is y q − y = x q i +1 − x − λ , with i apositive integer and gcd( r, p ) = 1, we determine the number of rational points in termsof Legendre symbol and p − adic valuation, given by the following theorem. Theorem 1.1.
Let i, r be integers such that < i < r , r = 2 b ˜ r and ˜ r = t a · · · t a u u where t j are distinct odd primes such that gcd( t j , p ) = 1 . For λ ∈ F q , the number of rational Mathematics Subject Classification.
Primary 12E20 Secondary 11T06.
Key words and phrases. circulant matrices, Artin Schreier curve, Hasse Weil bound, Irreducible poly-nomial with prescribed trace. points in F q r of the curve y q − y = x q i +1 − x − λ is q r − Dq ( r + L ) / where(i) D = ( − ( q − b − gcd(2 b ,i )) / and L = − ˜ r gcd(2 b , i ) , if b ≥ and i is even;(ii) D = Q uj =1 (cid:16) qt j (cid:17) max { ,ν tj ( r ) − ν tj ( i ) } and L = 2 b gcd(˜ r, i ) , if b = 0 , or b ≥ and i is odd.For λ = 0 the number of rational points in F q r of the curve y q − y = x q i +1 − x − λ is q r − Dq ( r + L ) / ( q − where(i) D = ( − ( q − b − gcd(2 b ,i )) / and L = − ˜ r gcd(2 b , i ) , if b ≥ and i is even;(ii) D = Q uj =1 (cid:16) qt j (cid:17) max { ,ν tj ( r ) − ν tj ( i ) } and L = 2 b gcd(˜ r, i ) , if b = 0 , or b ≥ and i is odd. When i = 1 we obtain the curve y q − y = x q +1 − x that is associated to the numberof monic irreducible polynomials with first two coefficients prescribed (see [3, 10]).The paper is organized as follows. The section 2 provides the background materialand important preliminary results. In Section 3 we discuss the case when C : y q − y = x · P ( x ) − λ with P ( x ) a F q -linearized polynomial. In Section 4 we given a explicitlyformula for the number of rational points in F q r for the curve y q − y = x q i +1 − x − λ and finally in Section 5 we given an alternative solution for the number of rational pointswhen i = 1 that includes the case when gcd( r, p ) = p .2. Preliminary results
Throughout this paper F q denotes a finite field with q elements, where q is a powerof an odd prime p . An important tool used in this section is the complete homogeneoussymmetric polynomials in r variables that are describe in the following definition. Definition 2.1.
The complete homogeneous symmetric polynomials of degree k , denotedfor h k ( r ) , is given by h k ( x , x , . . . , x r ) = X ≤ i ≤···≤ i k ≤ r x i · · · x i k . The following theorem, that can be found in [13] without prove, describe an anotherrepresentation for the polynomial h k . Theorem 2.2 ([13], Ex. 7 . . The complete homogeneous symmetric polynomial h k ( r ) ofdegree k in the variables x , x , . . . , x r can be expressed as the sum of rational functions h k ( x , . . . , x r ) = r X l =1 x r + k − l Q r m =1 m = l ( x l − x m ) . That type of polynomials will be useful to determining the rank of some circulantmatrices.
Definition 2.3.
Given a , a , . . . , a r − elements of a field L , let definea) The r × r circulant matrix C = ( c ij ) associated to the r -tuple ( a , a , . . . , a r − ) by therelation c ij = a k if j − i ≡ k (mod r ) . Let denote by C ( a , a , . . . , a r − ) this matrixand the vector ( a , a , . . . , a r − ) is called generator vector of C . N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 3 b) The associated polynomial of the circulant matrix C with generator vector ( a , a , . . . , a r − ) is f ( x ) = P r − i =0 a i x i . It is known(see for example [6]) that for a circulant matrix C = C ( a , a , . . . , a r − ) isvalid that det C = r Y i =1 ( a + a ω i + · · · + a r − ω r − i ) , where ω , . . . , ω r are the r -th roots of unity in some extension of F q in the case when r isrelatively prime with the characteristic p . We will use that fact in order to calculate therank of C . Specifically, the rank of C is related to the number of common roots of f ( x )and x r −
1. Before we determine the rank of C we need the following definitions. Definition 2.4.
For each < j ≤ k integers, A k and A k,j denote the polynomials(1) A k ( x , . . . , x k ) = Y ≤ t
Let j ≤ k be positive integers and h r,j ( k ) be the polynomial h r ( x , . . . , ˆ x j , . . . , x k ) , where ˆ x j means to omit the variable x j . Then k X j =1 x r − j A k,j h r − k +1 ,j ( k ) = 0 , for all k ≥ . Proof.
Let denote ǫ l,j = l > j − l < j l = j . By Theorem 2.2 it follows that k X i =1 x r − j A k,j h r − k +1 ,j ( k ) = k X j =1 x r − j ( − j +1 Y ≤ t
Lemma 2.6.
Let k ≥ be an integer and consider the polynomial F ( x , . . . , x k +1 ) = k +1 X j =1 x . . . x k +1 x j A k +1 ,j . Then F ( x , . . . , x k +1 ) = A k +1 . Proof.
We prove the result by induction on the number of variables. For k = 2 we have F ( x , x , x ) = X j =1 x x x x j ( − j +1 Y ≤ s 2. The polynomial F ( x , . . . , x k +1 ) has degree (cid:0) k +12 (cid:1) and if x i = x j , with 1 ≤ i < j ≤ k + 1, it follows that F ( x , . . . , x k +1 ) = 0, then A k +1 ( x , . . . , x k +1 ) divides F ( x , . . . , x k +1 ). Since A k +1 has the same degree of F weobtain that F ( x , . . . , x k +1 ) = c · A k +1 , for some c ∈ F q . It remains to show that c = 1.In order to prove that, we consider the coefficient of the monomial x kk +1 in the equation F ( x , . . . , x k +1 ) = c · A k +1 . Then we have this coefficient is k X j =1 x · · · x k x j ( − j +1 Y ≤ s For a straightforward calculation, dividing the result of Lemma 2.6 for A k +1 we obtain that k X j =1 x · · · x k x j Q k r =1 r = j ( x r − x j ) = 1 . Lemma 2.8. Let C be a r × r circulant matrix over F q , with generator vector ( a , a , · · · , a r − ) and f ( x ) be the associated polynomial to the matrix C , where gcd( r, p ) = 1 . Let g ( x ) = gcd( f ( x ) , x r − and α , α , . . . , α m the roots of g ( x ) . Then for each positiveinteger j ≤ m the relation ( a , a , . . . , a r − j ) · (1 , h ( α , . . . , α j ) , · · · , h r − j ( α , . . . , α j )) = 0 , (1) is satisfied, where · denotes the inner product.Proof. Let denote ~α k = ( α , . . . , α k +1 ). Since gcd( r, q ) = 1 it follows that α i = α j for all i = j. We procede by induction on the number of roots. For α a root of g we have a + a α + · · · + a r − α r − = 0 . Then ( a , a , . . . , a r − ) · (1 , α, . . . , α r − ) = 0 and that relation is equivalent to the firstcase of the induction. If j = 2 for the pair of roots α , α we have the relations ( a α + a α + · · · + a r − α r = 0 a α + a α + · · · + a r − α r = 0 . Subtracting these equations we obtain a ( α − α ) + a ( α − α ) + · · · + a r − ( α r − − α r − ) = 0 . N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 5 Since A = α − α = 0 it follows that0 = ( a , a , . . . , a r − ) · (cid:0) α − α , α − α , . . . , α r − − α r − (cid:1) = ( a , a , . . . , a r − ) · A (cid:18) , h ( α , α ) A , . . . , h r − ( α , α ) A (cid:19) , and that relation proved the case j = 2. Let suppose that the relation (1) is true for anychoice of k different roots of g . Let α , . . . , α k +1 be k + 1 roots of g . By the inductionhypotheses, we have k +1 equations of the form (1), where for each one we do not considerone of the roots, i.e., the j -th equation is given by( a , . . . , a r − k ) · (1 , h ,j ( ~α k +1 ) , . . . , h r − k,j ( ~α k +1 )) = 0 . (2)Multiplying the vector (1 , h ,j ( ~α k +1 ) , . . . , h r − k,j ( ~α k +1 )) for α r − j A k +1 ,j and addingthese vectors we get the vector u = k +1 X j =1 α r − j ( A k +1 ,j , A k +1 ,j h ,j ( ~α k +1 ) , . . . , A k +1 ,j h r − k,j ( ~α k +1 )) . By Lemma 2.5, the last coordinate of the vector u is k +1 X j =1 α r − j A k +1 ,j h r − k,j ( ~α k +1 ) = 0 . (3)Besides that, denoting α = α · · · α k +1 , the first coordinate of u is a k +1 X j =1 α r − j A k +1 ,j = a k +1 X j =1 α r − j (cid:16) ( − j +1 Y ≤ s 1, the l -th coordenate of u is equal to a l k +1 X j =1 α r − j A k +1 ,j h l,j ( ~α k +1 ) = a l k +1 X j =1 α r − j (cid:16) ( − j +1 Y ≤ s ≤ t ≤ k +1 s,t = j ( α t − α s ) k +1 X i =1 i = j α l + k − i Q k +1 m =1 m = i,j ( α i − α m ) (cid:17) = a l k +1 X j =1 k +1 X i =1 i = j (cid:0) α r − j α l + k − i ( − j +1 ( − k − i Y ≤ s ≤ t ≤ k +1 s,t = i,j ( α t − α s ) ǫ i,j (cid:1) = a l α k +1 X j =1 k +1 X i =1 i = j (cid:0) αα j α l + k − i ( − k + j − i +1 Y ≤ s ≤ t ≤ k +1 s,t = i,j ( α t − α s ) ǫ i,j (cid:1) . (5)Let us denote G ( x , . . . , x k +1 ) = k +1 X j =1 k +1 X i =1 i = j (cid:16) x · · · x k +1 x j x l + k − i ( − k + j − i +1 Y ≤ s ≤ t ≤ k +1 s,t = i,j ( x t − x s ) ǫ i,j (cid:17) . DANIELA OLIVEIRA F. E. BROCHERO MART´INEZ We observe that for x i = x j with i = j we have G ( x , . . . , x k +1 ) = 0, therefore ( x i − x j )divides G ( x , . . . , x k +1 ) for all i = j and then A k +1 ( x , . . . , x k +1 ) divides G ( x , . . . , x k +1 ),which can be expressed by the relation1 A k +1 G ( x , . . . , x k +1 ) = k +1 X i =1 x · · · x k +1 x l + k − i Q k +1 m =1 m = i ( x i − x m ) k +1 X j =1 j = i ( x i − x j ) Q k +1 r =1 r = i,j ( x r − x j )= k +1 X i =1 x l + ki Q k +1 m =1 m = i ( x i − x m ) k +1 X j =1 j = i x · · · x k +1 x i x j Q k +1 r =1 r = i,j ( x r − x j ) . Fixing i , it follows by Remark 2.7 that k +1 X j =1 j = i x · · · x k +1 x i x j Q k +1 r =1 r = i,j ( x r − x j ) = 1 . Therefore G ( x , · · · , x k +1 ) = A k +1 k +1 X i =1 x l + ki Q k +1 m =1 m = i ( x i − x m ) = A k +1 h l ( k + 1) . (6)By Equation (5) we have that a l k +1 X j =1 α r − j A k +1 ,j h l,j ( ~α k +1 ) = a l α A k +1 h l ( k + 1). Hence byEquation (3), (4) and (6) we conclude( a , . . . , a r − k − ) · (1 , h ( α , . . . , α k +1 ) , . . . , h n − k − ( α , . . . , α k +1 )) = 0 . (cid:3) Remark 2.9. For any λ root of g ( x ) and ≤ i ≤ m , multiplying f ( λ ) for λ i , it followsthat a r − i + a r − i +1 λ + · · · + a r − i − λ r − i = 0 and using Lemma 2.8 we have that results isstill true for any shift of the coefficients a , a , . . . , a r − . Remark 2.10. If g ( x ) has only simple roots then the result is also true in the case when gcd( r, q ) = 1 . The following theorem shows how to find an equivalent matrix to the circulant matrix C , but before we need the following definition. Definition 2.11. Let f ( x ) ∈ F q [ x ] be a monic polynomial of degree n such that f (0) = 0 .The reciprocal polynomial f ∗ of the polynomial f is defined by f ∗ ( x ) = f (0) x n f (cid:0) x (cid:1) . Thepolynomial f is self-reciprocal if f = f ∗ . Theorem 2.12. Let C be a r × r circulant matrix over F q and ( a , a , . . . , a r − ) thegenerator vector of C . Let f ( x ) be the polynomial associated to C such that g ( x ) =gcd( f ( x ) , x r − is a self-reciprocal polynomial with deg g ( x ) = m . Then the rank of C is l = r − m and there exists an invertible matrix M ∈ M r ( F q ) such that M AM T = (cid:18) R 00 0 (cid:19) , where R = ( r i,j ) is the l × l matrix defined by r ij = a ij for ≤ i, j ≤ l and M T denotesthe transpose matrix of M .Proof. Let α , . . . , α m be the roots of g ( x ). Let consider the matrices B i formed from theidentity matrix changing the entries of the r − i + 1-th row by1 , h ( α , . . . , α i ) , . . . , h r − i ( α , . . . , α i ) , , . . . , . N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 7 Observe that B = ··· ··· ... ... ··· ... ... ··· α ··· α r − α r − and since α and α − are roots of g , theproduct B AB T has last row and columns with null entries. Let us denote M = B m B m − · · · B B , then from Lemma 2.8 and Remark 2.9 we have that M AM T = (cid:18) R 00 0 (cid:19) , where the matrix R is the matrix C reduced to its first l rows and columns. (cid:3) Example 2.13. Let q = 27 and r = 7 . Let us denote the r -th cyclotomic polynomial by Q r . Since ord r q = 2 , Q r splits into monic irreducible polynomials over F q [ x ] of thesame degree . Let h a i = F ∗ where a has minimal polynomial x + 2 x + 1 . Then Q r ( x ) = ( x + 2 a x + 1)( x + (2 a + a + 2) x + 1)( x + (2 a + 2 a + 2) x + 1) . If f ( x ) =( x + 2 a x + 1)( x + (2 a + a + 2) x + 1)( x − a )= x + x ( a + 2) + x ( a + a + 1) + x (2 a + 1) + x ( a + 2 a ) + 2 a, then g ( x ) = gcd( f ( x ) , x r − 1) = ( x + 2 a x + 1)( x + (2 a + a + 2) x + 1)= x + x ( a + a + 2) + x (2 a + a ) + x ( a + a + 2) + 1 , is a self-reciprocal polynomial and the circulant matrix with associated polynomial f ( x ) is C = a a +2 a a +1 a + a +1 a +2 1 00 2 a a +2 a a +1 a + a +1 a +2 11 0 2 a a +2 a a +1 a + a +1 a +2 a +2 1 0 2 a a +2 a a +1 a + a +1 a + a +1 a +2 1 0 2 a a +2 a a +12 a +1 a + a +1 a +2 1 0 2 a a +2 aa +2 a a +1 a + a +1 a +2 1 0 2 a . By Theorem 2.12 the rank of C is and has reduced matrix C ′ = (cid:18) a a +2 a a +10 2 a a +2 a a (cid:19) . Inaddition, det C ′ = a + 12 a + a = a = 0 . The number of rational points of y q − y = x · P ( x ) − λ In this section, we associated the number of rational points of the curve y q − y = x · P ( x ) − λ with the trace function, defined below. Definition 3.1. For r a positive integer. Let define the trace function F q r F q by Tr F qr / F q ( x ) = x + x q + · · · + x q r − . For simplicity, unless you say otherwise, we denote the trace function of F q r by Tr . Let C be the curve defined by the equation C : y q − y = x · P ( x ) − λ, with λ ∈ F q and P ( x ) = P li =0 a l x q l a F q -linearized polynomial. We are interesting tofind the number of rational points in F q r of C . Appling the trace function we obtainTr( x · P ( x )) = rλ. (7)Therefore for each rational point ( x , y ) ∈ C ( F q r ) corresponds a solution of Tr( x ( P ( x )) = rλ . Reciprocally for each solution of Tr( x ( P ( x )) = rλ we have q rational points of C ( F ∗ q r ) DANIELA OLIVEIRA F. E. BROCHERO MART´INEZ of the form ( x , y j ) where the y j ’s are the solutions of equations y q − y = x P ( x ) − λ. Let us denote by N r the number of solutions of Tr F qr / F q ( x · P ( x )) = rλ . Then C r = q · N r . Let P denote the r × r permutation matrix P = ··· 00 0 1 ··· ... ... ... ... ... ... 11 0 0 ... and P l = P l for each l non-negative integer. The following proposition associated the F q -linearized polynomial P ( x ) with an appropriated circulant matrix. Proposition 3.2. Let P ( x ) = P li =0 a l x q l be a F q -linearized polynomial. For λ ∈ F q ,the number of solutions of Tr(( x · P ( x )) = rλ in F q r is equal to the number of solutions ~z = ( z , z , . . . , z n ) T ∈ F rq of the quadratic form ~z T A~z = rλ where A = P li =0 a i ( P i + P Ti ) . Proof. Let denote Γ = { β , . . . , β r } an arbitrary base of F q r over F q and N Γ = β β q ··· β qr − β β q ··· β qr − ... ... ... ... β r β qr ··· β qr − r . Since Γ is a basis, then N Γ is an invertible matrix and for x ∈ F q r we can write x = P rj =1 β j x i , with x , . . . , x r ∈ F q . The equation Tr( x · P ( x )) = rλ is equivalentto r − X j =0 x q j · P ( x ) q j = rλ. (8)From the fact that P ( x ) is a F q -linearized polynomial and trace is a F q -linear function, itis sufficient to consider monomials of the form x · x q l . Let us denote L l ( x ) = Tr( x · x q l ),then L l ( x ) = r − X j =0 x q j · ( x q l ) q j = r − X j =0 r X s =1 β s x s ! q j r X k =1 β k x k ! q j + l = r X s,k =1 r − X j =0 β q j s β q j + l k ! x s x l . Consequently L l ( x ) is a quadratic form that has the following symmetric representation L l ( x ) = ( x x · · · x r ) B l x x ... x r , where B l = 12 M Γ ( P l + P Tl ) M T Γ . Making the variable change ( z z · · · z r ) = ( x x · · · x r ) M Γ we get a equivalent systemwhich has the same number of solutions. Therefore˜ L l ( z ) = ( z z · · · z r ) (cid:20) 12 ( P l + P Tl ) (cid:21) z z ... z r Using equation (8) and definition of matrix A , the result follows. (cid:3) The following theorem, about number of solutions of a quadratic form, is well-knownand we present without proof. N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 9 Theorem 3.3 ([7], Theorems 6 . 26 and 6 . . Let Φ be a quadratic form over F q r , with q a power of an odd prime. Let ϕ be the bilinear symmetric form associated to Φ , v =dim(ker( ϕ )) and ˜Φ the reduced nondegenerate quadratic form equivalent to Φ . Let S λ := |{ x ∈ F q r | Φ( x ) = λ }| , ∆ be the determinant of the quadratic form ˜Φ and χ the quadraticcharacter of F q , then(i) If r + v is even: S λ = (cid:26) q r − + Dq ( r + v − / ( q − if λ = 0; q r − − Dq ( r + v − / if λ = 0 , (9) where D = χ (( − ( r − v ) / ∆) .(ii) If r + v is odd: S λ = (cid:26) q r − if λ = 0; q r − + Dq ( r + v − / if λ = 0 . (10) where D = χ (( − ( r − v − / λ ∆) .In particular D ∈ {− , } . It proved straightforward that Theorems 2.12, 3.3 combined with Preposition 3.2 pro-vide a proof of the following theorem. Theorem 3.4. Let P ( x ) = P li =0 a i x q i be a F q -linearized polynomial and f ( x ) = P li =0 a i x i be the associated polynomial of P ( x ) . Let suppose that g ( x ) = gcd( f ( x ) , x r − is aself-reciprocal polynomial of degree m . Let R be a matrix as defined in Theorem 2.12and a = det R . For each λ ∈ F q , the number of rational points in F q r of the curve y q − y = x · P ( x ) − λ is(1) q r + υ (2 rλ ) q ( r + m − / χ (( − ( r − m ) / a ) , if r − m is even.(2) q r + q ( r + m +1) / χ (( − ( r − m − / rλa ) , if r − m is odd, Example 3.5. Let q = 27 , r = 7 and f ( x ) , g ( x ) be the polynomials of Example 2.13.Put P ( x ) = x q + x q ( a + 2) + x q ( a + a + 1) + x q (2 a + 1) + x q ( a + 2 a ) + 2 ax thathas associated polynomial f ( x ) . Since r − m is odd and det C ′ = a , by Theorem 3.4 thenumber of rational points in F q r of the curve y q − y = x · P ( x ) − λ with λ ∈ F q is q + q χ ( λ ) . Therefore if λ is a square in F ∗ q the number of rational points is q + q and if λ is not asquare in F q the number of rational points is q − q . Besides that, if λ = 0 the numberof rational points is q . In the following section we consider some special polynomials P ( x ) and calculate ex-plicitly the value of D of Theorem 3.3.4. The number of rational points of y q − y = x · ( x q i − x ) − λ Throughout this section for a prime t we denote ν t ( n ) the t -adic valuation of n , i.e., themaximum power of t that divides n . For any integer a , (cid:0) at (cid:1) denotes the Legendre symbol.We will use that symbol in order to determine the number of rational points in F q r of thecurve y q − y = x q i +1 − x − λ. In previous section, the number of rational points in F q r of y q − y = x · P ( x ) − λ , with P ( x ) a F q -linearized polynomial and λ ∈ F q , it was associatedto the number of elements x ∈ F q r such that Tr( xP ( x )) = rλ . For Φ a quadratic form wedefine the symmetric bilinear form of Φ by ϕ ( x, y ) = Φ( x + y ) − Φ( x ) − Φ( y ). Dimensionof Φ is given by dimension of the subspace { x ∈ F q r : ϕ ( x, y ) = 0 for all y ∈ F q r } . Sincetrace function of xP ( x ) defines a quadratic form, in order to determine the number of solutions of Tr( xP ( x )) = rλ it is necessary to establish the dimension of the symmetricbilinear form associated to that quadratic form that it is described by next proposition. Proposition 4.1. Let < i < r be integers and P ( x ) = P ij =0 a j x q j a F q -linearizedpolynomial. Let Φ i ( x ) = Tr( x · P ( x )) be a quadratic form over F q r and ϕ i ( x, y ) thesymmetric bilinear form associated to Φ i . If a = 0 , then dim ker( ϕ i ) = deg (cid:16) gcd (cid:16) i X j =0 a j ( x j + x r − j ) , x r − (cid:17)(cid:17) . (11) Proof. In order to determine the dimension of the kernel of ϕ i it is sufficient calculate thedimension of the symmetric bilinear form ϕ ( x, y ), i.e.,dim { x ∈ F q r | ϕ ( x, y ) = 0 for all y ∈ F q r } . Then ϕ ( x, y ) = Tr (cid:16) i X j =0 a j ( x + y ) q j +1 − i X j =0 a j x q j +1 − i X j =0 a j y q j +1 (cid:17) = r − X l =0 (cid:0) i X j =0 a j ( x + y ) q j + l + q l − i X j =0 a j x q j + l + q l − i X j =0 a j y q j + l + q l (cid:1) = i X j =0 a j (cid:0) r − X l =0 x q j + l y q l − x q l y q j + l (cid:1) = i X j =0 a j (cid:0) r − X j =0 (( x q j + x q r − j ) y ) q l (cid:1) = i X j =0 a j Tr(( x q j + x q r − j ) y ) = Tr (cid:0) i X j =0 a j ( x q j + x q r − j ) y (cid:1) . By later equation, the relation ϕ ( x, y ) = 0 for all y ∈ F q r is equivalent to i X j =0 a j ( x q j + x q r − j ) = 0 . (12)The F q -linear subspace of F q r determined by Equation (12) is the set of roots of thepolynomial g ( x ) = gcd( h ( x ) , x q r − x ) where h ( x ) = P ij =0 a j ( x q j + x q r − j ). Since g is a F q -linearized polynomial, the degree of the associated polynomial given us the dimensionof ker( ϕ i ), that correspond to the degree of gcd( x r − , P ij =0 a j ( x j + x r − j )) and this factconcludes the proposition. (cid:3) The special case P ( x ) = x q i − x . For this case we explicitly determine the di-mension of the quadratic form. Besides that, we can use that information to count thenumber of rational points in F q r of the curve y q − y = x q i +1 − x − λ with λ ∈ F q . Thefollowing corollary is consequence of Proposition 4.1. Corollary 4.2. Let Φ i ( x ) = Tr( x q i +1 − x ) , with < i < r , be a quadratic form over F q r and ϕ i ( x, y ) the symmetric bilinear form associated to Φ i . Put r = p u ˜ r , i = p s ˜ i , where u, s are non negative integers such that gcd( p, ˜ r ) = gcd( p, ˜ i ) = 1 . Then dim ker( ϕ i ) = gcd(˜ r, ˜ i ) min( p u , p s ) . (13) N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 11 Proof. By Preposition 4.1 we need to determine the set of roots of the following polynomialgcd( x q i + x q r − i − x, x q r − x ) = gcd( x q i − x q i + x, x q r − x ) . (14)Since r = p u ˜ r , i = p s ˜ i , the associated polynomial to (14) isgcd (cid:0) x i − x + 1 , x r − (cid:1) = gcd(( x ˜ i − p s , ( x ˜ r − p u ) = ( x gcd(˜ r, ˜ i ) − min ( p u , p s ) . Therefore, from the fact that the degree of the later polynomial given us the dimensionof the kernel, it follows that dim ker( ϕ i ) = gcd(˜ r, ˜ i ) min( p u , p s ) . (cid:3) Using Theorem 3.3 and previous lemma we can determine the number of solutions ofTr( x q i +1 − x ) = rλ which will give us a complete description of the number of rationalspoints in F q r of the curve y q − y = x q i +1 − x − λ . Lemma 4.3. Let i, r be integers such that < i < r . Let Φ i ( x ) = Tr( x q i +1 − x ) be aquadratic form over F q r , where r = t a · · · t a u u and t j are distinct odd primes satisfying gcd( t i , p ) = 1 . Let v be the dimension of the bilinear symmetric form associated to Φ i and put i = p s ˜ i , where s is a non negative integer and gcd(˜ i, p ) = 1 . Then r + v is evenand, for λ ∈ F ∗ q , the constant D of Theorem 3.3 it is given by D = u Y j =1 (cid:18) qt j (cid:19) max { ,ν ti ( r ) − ν tj ( i ) } . Proof. The number of solutions S λ of Tr( x q i +1 − x ) = λ is given by Equation (9). Foreach λ ∈ F ∗ q , if Tr( x q i +1 − x ) = λ thenTr(( x q j ) q i +1 − ( x q j ) ) = Tr(( x q i +1 − x ) q j ) = Tr( x q i +1 − x ) = λ, (15)for all 0 ≤ j ≤ r − . We have two cases to consider1) r = t b , t an odd prime and gcd( t, p ) = 1.Using Equation (15), for each x ∈ S λ we can associated another d − d is the minimum positive divisor of r = t b such that x q d = x . We have that d > 1, because otherwise we would have x q = x and x q i +1 − x = x − x = 0 , whichmeans that λ = 0, a contradiction. Then d is a multiple of t and Equation (9) ofTheorem 3.3 module t can be rewritten as q r − − Dq ( r + v − / ≡ t ) , that is equivalent to D ≡ ( q ( r + v − / ) − ≡ q ( r + v − / (mod t ) , where in the last congruence we use that D = ± 1. By Lemma 4.2 we obtain D ≡ q ( t b + t min { b,νt ( i ) } ) / − (mod t ) ≡ q t min ( b,νt ( i )) ( t ( b − min ( b,νt ( i )) +1) / − (mod t ) ≡ q ( t ( b − min( s,νt ( i )) − / (mod t ) ≡ q ( t max { ,b − νt ( i ) } − / (mod t ) ≡ (cid:16) qt (cid:17) ( t max { ,b − νt ( i ) } − / ( t − (mod t ) Since (cid:0) qt (cid:1) assumes only values {− , } and t l − t − ≡ l (mod 2), we conclude D = (cid:16) qt (cid:17) max { ,b − ν t ( i ) } . 2) Now we consider the general case t = t a · · · t a u u , with t j being distinct odd primes sat-isfying gcd( t i , p ) = 1. We will prove the result by induction on the number of distinctprimes factors u of r . We already proved the case when u = 1. Let suppose that theresult is valid for u − ≥ r = t a · · · t a u u . By Lemma 4.2 thedimension of the bilinear symmetric form associated to Φ i ( x ) is v = gcd( t a · · · t a u u , i )and this fact implies that v divides t a · · · t a u u and r + v is even. Using Theorem 3.3for λ ∈ F ∗ q we have S λ = q r − − Dq ( r + v − / . Let λ be fixed and put r = ˜ rt a u u where ˜ r = t a · · · t a u − u − . For the subfield F q ˜ r ⊂ F q r ,applying induction hypothesis, the number of solutions of Tr F q ˜ r / F q ( x q i +1 − x ) = λ is S λ, ˜ r = q ˜ r − − u − Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } q (˜ r + v − / , where v is the dimension of kernel of the bilinear symmetric form associated toTr F q ˜ r / F q ( x q i +1 − x ). From Lemma 4.2 v = gcd(˜ r, i ) that implies v = v · gcd( t a u u , i ).Since F q ˜ r ∩ F q tauu = F q , the solutions that are not in F q ˜ r can be grouped in sets of sizecongruent to zero module t u , because if α is a solution then α q j also is a solution andsince α ∈ F q r it follows that there exists d > t a u u such that α q d = α . Then S λ ≡ S λ,m (mod t u ) , that is equivalent to q ˜ rt auu − − Dq (˜ rt auu + v − / ≡ q ˜ r − − u − Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } q (˜ r + v − / (mod t u ) . Using that q t u ≡ q (mod t u ), previous equation is equivalent to Dq (˜ rt auu + v − / ≡ u − Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } q (˜ r + v − / (mod t u ) . (16)Let v = gcd( p a u u , i ). We observe that q ( t auu − / ≡ (cid:16) qt u (cid:17) a u (mod t u ) and using thisrelation we obtain that q (˜ r + v − ˜ rt auu − v ) / ≡ q − ˜ r (cid:16) tauu − (cid:17) q ( v − v ) (mod t u ) ≡ (cid:18) qt u (cid:19) a u ˜ r q ( v − v ) (mod t u ) ≡ (cid:18) qt u (cid:19) a u ˜ r q v − v (mod t u ) ≡ (cid:18) qt u (cid:19) a u q − v p min { au,νtu ( i ) } u − (mod t u ) . (17) N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 13 Equations (16) and (17) allow us to conclude that D ≡ u − Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } q (˜ r + v − ˜ rt auu − v ) / (mod t u ) ≡ u − Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } (cid:18) qt u (cid:19) a u q − v t min { au,νtu ( i ) } u − (mod t u ) ≡ u − Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } (cid:18) qt u (cid:19) a u (cid:18) qt u (cid:19) − min { a u ,ν tu ( i ) } (mod t u ) ≡ u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } (mod t u ) . Consequently D = u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } . (cid:3) Remark 4.4. The case when λ = 0 , follows using Theorem 3.3 since D is the samefor any λ ∈ F q . Then by Lemma 4.3 we have for λ = 0 , the D of equation (9) is Q uj =1 (cid:16) qt j (cid:17) max { ,ν ti ( r ) − ν tj ( i ) } . For extensions of degree power of 2, we have the following result. Lemma 4.5. Let b, i, r be integers such that < i < r and r = 2 b . Let Φ i ( x ) = Tr( x q i +1 − x ) be a quadratic form over F q r and v be the dimension of the bilinear symmetric formassociated to Φ i . For any λ ∈ F ∗ q it follows that the D defined in Theorem 3.3 is given by(i) If b = 1 , then D = χ ( − λ ); (ii) If b ≥ and r + v is even, then D = ( − ( q − b − v ) / .(iii) If b = 2 and r + v is odd, then D = ( − ( q +1) / · χ (cid:0) − λ b − (cid:1) .And for λ = 0 we have(i) If b = 1 : D = χ ( − λ ); (ii) If b ≥ and r + v is even : D = ( − ( q − b − v ) / ;(iii) If b ≥ and r + v is odd: D = 0 . Proof. When r = 2 it follows that i = 1 and Tr( x q +1 − x ) = λ, where x ∈ F q , that isequivalent to x q + q − x q + x q +1 − x = λ (18)and it can be written as ( x q − x ) = − λ . If λ = 0 that relation is equivalent to x q − x = 0and therefore x ∈ F q . In that case we have q solutions. For λ ∈ F ∗ q , let us consider thefollowing maps ψ : F q → F q τ : F q → F q x x q − x x x In order to determine the number of solutions of Equation (18) it is enough prescribe all x ∈ F q such that τ ( ψ ( x )) = − λ . Let { , α } be a base of F q / F q . The image of { , α } by ψ is { , β } , where β = α q − α . Since ker( ψ ) = F q the image of ψ is generated by β .Therefore it is enough to consider elements of the form x = cα with c ∈ F q . That is τ ( ψ ( cα )) = τ ( cβ ) = c β . Claim: β / ∈ F q . Let suppose, by contradiction, that β q = β . Then α q − α q − α q + α = 0 , that only occurs if − α q − α ) = 0. But that later relation is not possible, because α ∈ F q \ F q and p = 2. Therefore τ ( ψ ( x )) = − λ if and only if c β = − λ and since β / ∈ F q that equation has solution if and only if − λ is not a square in F q . In this case wehave 2 q solutions for Equation (18).Now we consider the case when r = 2 b with b > 1. Let i = p s ˜ i , with gcd( p, ˜ i ) = 1.From Lemma 4.2 v = gcd(2 b , ˜ i ) min (1 , p s ) = gcd(2 b , i ), then v is of the form 2 c with0 ≤ c ≤ b . We divide the proof in two cases.(1) r + v is even.In this case v and i are even. The number of solutions of Tr( x q i +1 − x ) = λ is given by Equation (9). If Tr( α q i +1 − α ) = λ , for some α ∈ F ∗ q , thenTr(( α q j ) q i +1 − ( α q j ) ) = Tr(( α q i +1 − α ) q j ) = Tr( α q i +1 − α ) = λ, (19)for each 0 ≤ j ≤ r − 1. Since r = 2 b ≥ x ∈ S λ wecan associated another d − d is the minimum positive divisorof r = t b such that α q d = α . We claim that d > 2. In fact if d = 1 then( α q i +1 − α ) q = α − α = 0 that implies λ = 0, a contradiction. In the casewhen d = 2, for α ∈ F q it follows that ( α q i +1 − α ) q = α q i +1 − α = α − α = 0,because i is even. That relation also implies λ = 0, that is also a contradiction.Then Equation (19) does not have solutions in that cases. Consequently d > d is multiple of 4 and Equation (9) of Theorem 3.3 module 4 is q b − − Dq (2 b + v − / ≡ , that is equivalent to D ≡ q b − − (2 b + v − / ≡ q (2 b − v ) / (mod 4) . We conclude that D = ( − ( q − b − v ) / .(2) r + v is odd.In this case v is odd and divisor of 2 b , then v = 1. Using the same argumentas in the previous case, the number of solutions of Tr( x q i +1 − x ) = λ is given byEquation (10). Besides that, for λ ∈ F ∗ q , it follows that Tr( x q i +1 − x ) = λ if andonly ifTr(( α q j ) q i +1 − ( α q j ) ) = Tr(( α q i +1 − α ) q j ) = Tr( α q i +1 − α ) = λ, (20)for all 0 ≤ j ≤ r − . Since r = 2 b ≥ x ∈ S λ wecan associated another d − d is the minimum divisor of r = t b such that α q d = α . The case d = 1 can not occurs, otherwise we would have λ = 0.For x ∈ F q ⊂ F q r , we have λ = Tr( x q i +1 − x ) = 2 b − · (( x q i +1 + q − x q ) + x q i +1 − x ) . (21)For the same reason as in previous case, Equation (21) does not have solution in F q . Therefore we can suppose that x ∈ F q \ F q and equation λ = 2 b − · ( x q +1 + x q +1 − x q − x ) N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 15 can be written as ( x q − x ) = − γ, where γ = λ b − . Using the same argument ofitem (i) of Lemma 4.5 it follows that ( x q − x ) = − γ has solutions if and onlyif − γ is not a square in F q , that is equivalent to − λ b − not be a square in F q andin this case we have 2 q solutions for Equation (18). Consequently the number ofsolutions of Equation (21) is (cid:0) − χ ( − λ b − ) (cid:1) q . Then S λ − (cid:18) − χ (cid:18) − λ b − (cid:19)(cid:19) q ≡ . Now by Theorem 3.3 it follows that q b − + Dq (2 b + v − / ≡ (cid:18) − χ (cid:18) − λ b − (cid:19)(cid:19) q (mod 4) , that is equivalent to D ≡ (cid:0) − χ (cid:0) − λ b − (cid:1)(cid:1) q − (2 b + v − / − q (2 b − v − / (mod 4) . Therefore D ≡ (cid:18)(cid:18) − χ (cid:18) − λ b − (cid:19)(cid:19) q − b − (cid:19) q (2 b − v − / (mod 4) . Since v = 1 and q ≡ D ≡ − q b − − · χ (cid:18) − λ b − (cid:19) ≡ − q · χ (cid:18) − λ b − (cid:19) (mod 4) . Consequently D = ( − ( q +1) / · χ (cid:0) − λ b − (cid:1) . The case when λ = 0 follows by Theorem 3.3, that assure us that D is the same for λ equal or different to zero if r + v is even and when r + v is odd, D = 0. (cid:3) Using Theorem 3.3 and Lemma 4.5 we determine the value of S λ . Theorem 4.6. Let b, i, r be integers such that < i < r and r = 2 b . Let Φ i ( x ) =Tr( x q i +1 − x ) be a quadratic form over F q r and v be the dimension of the bilinear sym-metric form associated to Φ i . For λ ∈ F ∗ q , the number of solutions S λ of Φ i ( x ) = λ in F q r is given by(i) If b = 1 : S λ = (1 − χ ( − λ )) q ; (ii) If b ≥ and r + v is even : S λ = q b − − ( − ( q − b − v ) / q (2 b + v − / ;(iii) If b ≥ and r + v is odd: S λ = q b − − ( − ( q − / · χ (cid:0) − λ b − (cid:1) q (2 b + v − / . For λ = 0 , the number of solutions S λ of Φ i ( x ) = λ in F q r is given by(i) If b = 1 : S = (1 − χ ( − λ )) q ; (ii) If b ≥ and r + v is even : S = q b − − ( − ( q − b − v ) / q (2 b + v − / ( q − ;(iii) If b ≥ and r + v is odd: S = q b − . The results obtained in Lemmas 4.3, 4.6 can be used inductively to obtain the followingresult for extensions of degree r with gcd( r, p ) = 1. Theorem 4.7. Let b, i, r be integers such that < i < r and r = 2 b ˜ r, r = t a · · · t a u u where t j are distinct odd primes such that gcd( t j , p ) = 1 . Let Φ i ( x ) = Tr( x q i +1 − x ) be thequadratic form over F q r and v the dimension of the bilinear symmetric form associatedto Φ i . For λ ∈ F ∗ q ,(i) S λ = q r − − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj (˜ r ) − ν tj ( i ) } q r +2 v − , if i is even and b = 1 .(ii) S λ = q r − − ( − ( q ˜ r − b − v ) / q ( r − ˜ rv − , if i is even and b ≥ . (iii) S λ = q r − − Q uj =1 (cid:16) qt j (cid:17) max { ,ν tj ( r ) − ν tj ( i ) } q ( r +2 bv − , if i is odd.where v = gcd(˜ r, i ) and v = gcd(2 b , i ) .For λ = 0 , the number of solutions S λ of Φ i ( x ) = λ in F q r is given by(i) S = q r − − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj (˜ r ) − ν tj ( i ) } q r +2 v − ( q − , if i is even and b = 1 .(ii) S = q r − − ( − ( q ˜ r − b − v ) / q ( r − ˜ rv − ( q − , if i is even and b ≥ .(iii) S = q r − if i is odd.where v = gcd(˜ r, i ) and v = gcd(2 b , i ) .Proof. By Lemma 4.2 it follows that v = gcd( r, i ).(i) Using the relation of transitivity of the trace function we obtain λ = Tr( x q i +1 − x ) = Tr F q / F q (Tr F qr / F q ( x q i +1 − x )) . (22)Let µ ∈ F ∗ q ˜ r such that µ = Tr F qr / F q ( x q i +1 − x ). Then Equation (23) is equivalentto the system ( Tr F q / F q ( µ ) = λ ;Tr F qr / F q ( x q i +1 − x ) = µ. Put Q = q and then Q ˜ r = q r . Since b = 1, by Lemma 4.3 it follows that thenumber of solutions of Tr F qr / F q ˜ r ( x q i +1 − x )) = µ is Q ˜ r − − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj (˜ r ) − ν tj ( i ) } Q ˜ r + v − . The dimension of the kernel of the quadratic form Tr F q / F q is v = gcd(˜ r, i ). Thenumber of solutions of Tr F q ˜ r / F q ( µ ) = λ is q , then for λ ∈ F ∗ q S λ = q Q ˜ r − − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj (˜ r ) − ν tj ( i ) } Q ˜ r + v − = q q r − − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj (˜ r ) − ν tj ( i ) } q r +2 v − = q r − − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj (˜ r ) − ν tj ( i ) } q r +2 v − . (ii) Using the relation of transitivity of the trace function we obtain λ = Tr( x q i +1 − x ) = Tr F q ˜ r / F q (Tr F qr / F q ˜ r ( x q i +1 − x )) . (23)Let µ ∈ F ∗ q ˜ r such that µ = Tr F q ˜ r / F q ( x q i +1 − x ). Then Equation (23) is equivalentto the system ( Tr F q ˜ r / F q ( µ ) = λ ;Tr F qr / F q ˜ r ( x q i +1 − x ) = µ. N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 17 Put Q = q ˜ r and then Q b = q r . Since b ≥ 2, from item (ii) of Lemma 4.6 it followsthat the number of solutions of Tr F qr / F q ˜ r ( x q i +1 − x )) = µ is S µ = Q b − − ( − ( Q − (2 b − v Q (2 b − v − , where v = gcd(2 b , i ) is the dimension of the kernel of the quadratic form Tr F qr / F q ˜ r ( x q i +1 − x ). Besides that, the number of solutions of Tr F q ˜ r / F q ( µ ) = λ is q ˜ r − , then the num-ber of solutions of Φ i ( x ) = λ is q ˜ r − ( Q b − − ( − ( Q − (2 b − v Q (2 b − v − ) = q ˜ r − ( q r − ˜ r − ( − q (˜ r − 1) (2 b − v q ( r − ˜ rv − r )2 )= q r − − ( − q (˜ r − 1) (2 b − v q ( r − ˜ rv − . (iii) Put Q = q b and then Q ˜ r = q r . Using the relation of transitivity of the tracefunction we have λ = Tr( x q i +1 − x ) = Tr F Q / F q (Tr F qr / F Q ( x q i +1 − x )) . (24)Let µ ∈ F ∗ Q such that µ = Tr F Q / F q ( x q i +1 − x ). The number of solutions of (24) isequal to the number of solutions of the system (cid:26) Tr F Q / F q ( µ ) = λ ;Tr F qr / F Q ( x q i +1 − x )) = µ. By Corollary 4.3 we have that the number of solutions of Tr F qr / F Q ( x q i +1 − x )) = µ is Q ˜ r − − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } Q (˜ r + v − , where v = gcd(˜ r, i ) is the dimension of the kernel of the quadratic form Tr F qr / F Q ( x q i +1 − x )). Since the number of solutions of Tr F Q / F q ( µ ) = λ is q b − we conclude that thenumber of solutions of Φ i ( x ) = λ is S λ = q b − Q ˜ r − − u Y j =1 (cid:18) Qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } Q (˜ r + v − = q b − q r − b − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } q ( r +2 bv − b +1)2 = q r − − u Y j =1 (cid:18) qt j (cid:19) max { ,ν tj ( r ) − ν tj ( i ) } q ( r +2 bv − . The case when λ = 0 following using the same ideas e the corresponding formulas for S in extensions of degree a power of 2 or of degree odd. (cid:3) Using Lemma 4.3 and Theorem 4.7 we can determine the number of rational points ofthe curve y q − y = x q i +1 − x − λ , that proves Theorem 1.1.In the following section, we calculate the number of solutions of the curve y q − y = x q i +1 − x − λ, the particular case when i = 1. We use other method that allows us tosolve the case when extension of degree r is such that gcd( r, p ) = p. The particular case y q − y = x ( x q − x ) − λ In this section { β , . . . , β r } denote a base of F q r over F q and B = β β q · · · β q r − β β q · · · β q r − ... · · · . . . ... β r β q · · · β q r − r . We use that matrix as a tool to associate to the number of rational points of y q − y = x ( x q − x ) − λ with the number of x ∈ F q r such thatTr( f ( x )) = rλ. (25)Let us denote N r by the number of solutions of Tr( f ( x )) = 0 , for x ∈ F q r , since we saw inprevious section, the number C r of solutions for y q − y = x ( x q − x ) − λ in F q r it is givenby C r = q · N r . The following proposition associated Tr( x q +1 − x − λ ) with quadraticforms. Proposition 5.1. Let f ( x ) = x q +1 − x − λ, with λ ∈ F q . The number of solutions of Tr( f ( x )) = rλ over F q r is equal to the number of solutions of the quadratic form (cid:0) x x · · · x r (cid:1) A x x ... x r = rλ, where A = ( a j,l ) j,l , with a j,l = Tr( β qj β l + β ql β j − β j β l ) . Proof. Put x = P rj =1 β j x i with x i ∈ F q . The equation (25) is equivalent to r − X i =0 f ( x ) q i = rλ. (26)Reordering the terms r − X i =0 f ( x ) q i = r − X i =0 (cid:16) r X j =1 β j x j (cid:17) q i (cid:16) r X l =1 ( β ql − β l ) x l (cid:17) q i − rλ = r X j,l =1 (cid:16) r − X i =0 β q i j ( β q i +1 l − β q i l ) (cid:17) x j x l − rλ. The equation (26) assures us that P rj,l =1 (cid:16)P r − i =0 β q i j ( β q i +1 l − β q i l ) (cid:17) x j x l = rλ. Denoting ( a jl ) j,l = r − X i =0 β q i j β q i l ( β q i +1 − q i l + β q i +1 − q i j − a j,l = 12 Tr( β qj β l + β ql β j − β j β l ) , the result follows. (cid:3) The matrix A can be rewritten as A = 12 ( A + A + A ) , where A = (Tr( β qj β l )) j,l , A = (Tr( β j β ql )) j,l and A = (Tr( β j β l )) j,l . Using the per-mutation matrix P and the fact that P − = P T then A = B P T B T , A = B P B T and A = BB T . It follows that A = B ( P T − Id + P ) B T since B is an invertible matrix, in N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 19 order to determine the number of solutions of the quadratic form with matrix A we needto determine the rank of the matrix P T + P − Id . The following proposition determinesthat rank. Proposition 5.2. The rank of the matrix N = P T − Id + P over F q isrank N = (cid:26) r − if gcd( r, p ) = 1; r − if gcd( r, p ) = p. Proof. The matrix N ∈ M r ( F q ) is given by N = − ... − ... − ... ... ... ... ... ... ... ... ... − ... − ... − . Substitutingthe last row and column for the sum of all rows and then for the sum of all columns, weobtain − ... − ... − ... ... ... ... ... ... ... ... ... − ... − ... . Replacing the ( r − · · · + ( r − r − r − times the ( r − r − − ... − ... − ... ... ... ... ... ... ... ... ... ... − ... r 00 0 0 ... . Now the idea is the same as before but applying in the columns of the matrix, i.e.,summing the columns to obtain a matrix with the ( r − − ... − ... − ... ... ... ... ... ... ... ... ... ... − ... r 00 0 0 ... . Now we consider the matrix N r − , obtained after the first operation without the lastcolumn and row, that is N r − = − ... − ... − ... ... ... ... ... ... ... ... ... − ... − ... − ∈ M r − ( F q ) . Claim 1. Let L r − = det N r − , then L r − = ( − ( r − r. Expanding the determinant of N r − by the first row we have L r − = − L r − − L r − . This implies the recurrent relation L r − + 2 L r − + L r − = 0, that have as characteristicpolynomial λ + 2 λ + 1 = ( λ + 1) , whose double root is − 1. Therefore the solution forthe recurrent relation is L r − = A ( − r + B ( − r r , where A, B ∈ F q . Since L = 3e L = − 4, we conclude that A = 0 e B = − L ( r − = ( − ( r − r concluding the claim.Besides that, if gcd( r, p ) = 1 it follows that L r − = ( − ( r − r = 0 and the rank of N r − is r − N is r − In the case where gcd( r, p ) = p , we consider N r − the reduced matrix of − ... − ... − ... ... ... ... ... ... ... ... ... − ... − ... r ∈ M r − ( F q )obtained without the last row and column. Therefore L r − = ( − r − ( r − = 0 and therank of N is r − N = (cid:26) r − r, p ) = 1; r − r, p ) = p. (cid:3) By Theorem 3.3 and Propositions 5.1 and 5.2 we have the following theorem. Theorem 5.3. Let λ ∈ F q and r a positive integer. The number of rational points in F q r of the curve y q − y = x q +1 − x − λ , denoted by C r , is C r = q r + q ( r +2) / χ (( − r/ λ ) if gcd( r, p ) = 1 and r is even; q r + q ( r +1) / ν (2 rλ ) χ (( − ( r − / r ) if gcd( r, p ) = 1 and r is odd; q r + q ( r − / ν (2 rλ ) χ (( − ( r − / ) if gcd( r, p ) = p and r is even; q r if gcd( r, p ) = p and r is odd. Some open problems We finished this paper enumerating some open problems. We note that Theorem 3.4determine the number of rational points in F q r of the curve C : y q − y = x · P ( x ) − λ when P ( x ) is a F q linearized polynomial and satisfies the condition that g ( x ) = gcd( f ( x ) , x r − f ( x ) is the associated polynomial of P ( x ). Fromthat we have two problems: Problem 1. Determine the number of rational points of C when P ( x ) is not a F q -linearized polynomial. Problem 2. Determine the number of rational points of C when g ( x ) is not a self-reciprocal polynomial. In Section 4, we only study extensions of degree r such that gcd( r, p ) = 1. Whathappens in the general case? That is: Problem 3. Using similar methods or others, determine explicitly the number of rationalpoints in F q r of the curve y q − y = x · P ( x ) − λ , where λ ∈ F q and P ( x ) is a F q linearizedpolynomial, such that gcd( r, p ) = p Problem 4. Determine the number of solutions of y q − y = P ( x ) Q ( x ) when P ( x ) , Q ( x ) ∈ F q [ x ] are F q -linearized polynomials. In [9], the authors show that the number of monic irreducible polynomials in F q [ x ] ofdegree r and with the first and third coefficients prescribed is related to the curve y q − y = x q +1 − x q +2 . Then, we have the following problem. Problem 5. Determine the number of solution of the curve y q − y = xP ( x ) Q ( x ) when P ( x ) , Q ( x ) ∈ F q [ x ] are F q -linearized polynomials. N CIRCULANT MATRICES AND RATIONAL POINTS OF ARTIN SCHREIER CURVES 21 References [1] Anbar, N. and Meidl, W. More on quadratic functions and maximal Artin-Schreier curves. ApplicableAlgebra in Engineering, Communication and Computing (5) 409-426, (2015).[2] Anbar, N. and Meidl, W. Quadratic functions and maximal Artin-Schreier curves Finite Fields Appl. The number of irreducible polynomialsover GF(2) with given trace and subtrace. J. Combin. Math. Combin. Comput., , 31 – 64 (2003).[4] Coulter R.S. The number of rational points of a class of Artin–Schreier curves. Finite Fields Appl. , 397-413 (2002).[5] Hefez, A. and Kakuta, N. Polars of Artin–Schreier curves Acta Arith. , 57-70 (1996).[6] Kra, I.; Simanca S. R. On Circulant Matrices Notices of the AMS, (2012).[7] Lidl, R.; Niederreiter, H. Finite Fields . Encyclopedia Math. Appl., Vol. , Addison-Wesley, Reading,MA, (1983).[8] Niederreiter,H., Xing, Ch. Rational Points on Curves Over Finite Fields: Theory and Applications Cambridge Uni. Press (2001)[9] Lal´ın, Matilde and Larocque, Olivier The number of irreducible polynomials with the first two pre-scribed coefficients over a finite field. Rocky Mountain J. Math. no. 5, 1587-1618 (2016).[10] McGuire, G. and Yılmaz, E. S. The Number of Irreducible Polynomials with the First Two Coeffi-cients Fixed over Finite Fields of Odd Characteristic Arxiv ID: 1609.02314.[11] Ozbudak F., Saygı Z.: Rational points of the curve y q n − y = γx q h +1 − α. over F q m AppliedAlgebra and Number Theory. Cambridge University Press, (2014).[12] Ozbudak, F. and Saygi, Z. Explicit maximal and minimal curves over finite fields of odd charac-teristics Finite Fields and Their App., Volume , 81-92 (2016).[13] Stanley, Richard P. Enumerative Combinatorics Cambridge University Press, Vol , (1999).[14] Tsfasman, M,. Vl˘adut¸, S., Nogin, D. Algebraic geometric codes: basic notions. Mathematical Surveysand Monographs, 139. AMS (2007).[15] Van der Geer, G. and Van der Vlugt, M. Fibre products of Artin-Schreier curves and generalizedHamming weights of codes J. Combin. Theory Ser. A70