Bivariate trinomials over finite fields
aa r X i v : . [ m a t h . N T ] F e b Bivariate trinomials over finite fields
M. Avenda˜no and J. Mart´ın-Morales ∗ February 23, 2021
Abstract
We study the number of points in the family of plane curves defined by a trinomial C ( α, β ) = { ( x, y ) ∈ F q : αx a y a + βx a y a = x a y a } with fixed exponents (not collinear) and varying coefficients over finite fields. We prove that each ofthese curves has an almost predictable number of points, given by a closed formula that depends on thecoefficients, exponents, and the field, with a small error term N ( α, β ) that is bounded in absolute valueby 2˜ gq / , where ˜ g is a constant that depends only on the exponents and the field. A formula for ˜ g isprovided, as well as a comparison of ˜ g with the genus g of the projective closure of the curve over F q . Wealso give several linear and quadratic identities for the numbers N ( α, β ) that are strong enough to provethe estimate above, and in some cases, to characterize them completely. The main result in this article is inspired by Theorem 1.1 given below, proven by Gauss in his bookDisquisitiones Arithmeticae [5, Thm. 358]. We have used a mildly rephrased version of the original theorem,taken from [9, page 111], that better matches our more modern notation.
Theorem 1.1 (Gauss) . Let p be an odd prime and let M p be the number of points in the projective curve { [ x : y : z ] ∈ P ( F p ) : x + y + z = 0 } . (a) If p , then M p = p + 1 . (b) If p ≡ , then the equation u + 27¯ v = 4 p has a unique integer solution (up to the signs),and if u is chosen such that u ≡ , then M p = p + 1 + u . In a few words, Gauss’ theorem says that the number of (projective) points in the plane curve x + y + z ≡ p ) is p + 1 plus a small error term u (that only appears when p ≡ u +27¯ v = 4 p with integral unknowns. Our main result (Thm. 2.2) is a generalizationof Gauss’ theorem for any non-degenerate trinomial equation in two variables, over any finite field, wherewe show that the number of points is a predictable number (given by a closed formula in terms of thecoefficients, exponents, and the field) plus an error term which also satisfies an explicit quadratic equationin many unknowns, all of them having a precise meaning (as opposed to Gauss’ theorem, where only thevariable u matters). More precisely, our result gives, in the case p ≡ M p = p + 1 + u , where u + v + uv = 3 p for some u, v ∈ Z . The symmetry of the curve allows one to rewrite it as u + 27¯ v = 4 p ,where ¯ v = v + u ∈ Z and to show that u ≡ u is bounded in absolute value by 2 √ p . This obser-vation was generalized by Hasse to elliptic curves over finite fields [8, Ch. 5, Thm. 1.1], then by Weil tohypersurfaces defined by an equation of the type α x a + α x a + · · · + α r x a r r = b [10], which led to thestatement of the famous Weil’s conjectures, finally proven by Dwork [4], Grothendieck [6], and Deligne [3]for any smooth hypersurface.With our approach, the estimate of the error follows from a simple computation using Lagrange multipliers(see Prop. 3.4). In contrast with the results above, our proof is elementary and the estimate is valid for anytrinomial (not necessarily smooth). Moreover, our estimate 2˜ gq / (see Cor. 2.3) is better that the bound ∗ Partially supported by the Spanish Government MTM2016-76868-C2-2-P,
Grupo E15 Gobierno de Arag´on/Fondo SocialEuropeo , and FQM-333 from
Junta de Andaluc´ıa . gq / , since the genus g is an invariant that only reflects the complexgeometry of the curve, while our ˜ g includes also information about the field. In Section 5, we obtain a closedformula for the genus g of a trinomial plane curve (see Prop. 5.2), that can be compared term by term withthe definition of ˜ g given in (1). For instance, in the case of Gauss’ theorem, the curve has genus g = 1, butour ˜ g is zero when p R . Some experiments show that a much better estimate could becomputed if we were able to solve the optimization problem over the integers (see Example 6.3). Let p be a prime and q = p n for some n ≥
1. Let ρ be a generator of the cyclic group F ∗ q . Consider the curve C ij = { ( x, y ) ∈ F q : ρ i x a y a + ρ j x a y a = x a y a } , and let C ∗ ij = C ij ∩ ( F ∗ q ) .To avoid a degenerate case, we assume that the exponents vectors ( a , a ), ( a , a ), ( a , a ) are notcollinear, i.e. the matrix B = (cid:20) b b b b (cid:21) := (cid:20) a − a a − a a − a a − a (cid:21) is invertible.We need the following constants derived from B : d = gcd( b , b , q − ,e = gcd( b , b , q − ,f = gcd( b − b , b − b , q − ,k = gcd(( q −
1) gcd( d, e, f ) , det( B )) ,w = ( q even , q − q odd , ˜ g = 12 ( k − d − e − f + 2) . (1)The value k corresponds to | coker( B ) | , where B is regarded as a group homomorphism B : Z q − → Z q − given by the multiplication v Bv (see Lemma 3.3).Our goal is to estimate the number of points |C ij | and |C ∗ ij | for all i, j . Since ρ q − = 1, the indices i and j can be regarded modulo q − Definition 2.1. D ℓ ( i ) = ( ℓ if ℓ | i, |C ij | = |C ∗ ij | + |C ij ∩ { x = 0 , y = 0 }| + |C ij ∩ { y = 0 , x = 0 }| + |C ij ∩ { x = y = 0 }| , and that thepoints in C ij ∩ { x = 0 , y = 0 } and C ij ∩ { y = 0 , x = 0 } correspond to the solutions in F ∗ q of a univariateequation with at most two non-zero terms. Therefore, |C ij ∩ { x = 0 , y = 0 }| and |C ij ∩ { y = 0 , x = 0 }| can becomputed exactly with a closed formula in terms of i , j , q , and the exponents (see Lemma 3.2). Moreover, |C ij ∩ { x = y = 0 }| is either 1 or 0, depending on whether a + a , a + a , and a + a are all positiveor not. This means that |C ij | and |C ∗ ij | can be easily derived from each other. For this reason, and to avoiddiscussing several cases depending on the configuration of the exponents, we present our results only for |C ∗ ij | , which can be done with a more uniform notation. Theorem 2.2.
With the notation given above, we have |C ∗ ij | = q + 1 − D d ( i ) − D e ( j ) − D f ( i − j + w ) + N ij (2) for some integers N ij that satisfy: (a) q − X j =0 N ij = 0 for all i , q − X i =0 N ij = 0 for all j , (c) X i − j = r N ij = 0 for all r , (d) N i + b ,j + b = N ij = N i + b ,j + b for all i, j , (e) q − X i =0 q − X j =0 N ij = 2˜ g ( q − q = ( q − q ( k − d − e − f + 2) . Using (d), the sum of Theorem 2.2(e) can be rewritten taking only one representative of each ( i, j ) modulothe subgroup h ( b , b ) , ( b , b ) i ⊆ Z q − , X ( i,j ) ∈ coker( B ) N ij = 2˜ gkq = kq ( k − d − e − f + 2) ≤ k q. (3)We immediately obtain the upper bound | N ij | ≤ k √ q for all i, j . Using a similar approach, but takingadvantage of (a), (b), and (c), it is possible to get a stronger upper bound: Corollary 2.3. | N ij | ≤ g √ q for all i, j . Lemma 3.1.
For any r ≥ , q − X i =0 D ℓ ( i ) r = ℓ r − ( q − . Proof.
By definition of D ℓ we have: q − X i =0 D ℓ ( i ) r = X ℓ | i ℓ r = ℓ r · q − ℓ = ℓ r − ( q − , since the number of indices 0 ≤ i < q − ℓ is exactly q − ℓ . Lemma 3.2.
For any a , . . . , a m ∈ Z , (cid:12)(cid:12)(cid:8) ( x , . . . , x m ) ∈ ( F ∗ q ) m : ρ i x a · · · x a m m = 1 (cid:9)(cid:12)(cid:12) = ( q − m − D ℓ ( i ) , where ℓ = gcd( a , . . . , a m , q − .Proof. Consider the group homomorphism ϕ : ( F ∗ q ) m → F ∗ q given by ( x , . . . , x m ) x a · · · x a m m . Theimage of ϕ is generated by ρ a , . . . , ρ a m , which is also generated by ρ ℓ since the group F ∗ q is cyclic, and inparticular | im( ϕ ) | = q − ℓ . When ρ − i
6∈ h ρ ℓ i , i.e. ℓ ∤ i , the left-hand side and the right-hand side of theequation in the statement are both clearly zero. Otherwise, when ℓ | i , the number of solutions is equal to | coker( ϕ ) | = ( q − m / | im( ϕ ) | = ( q − m − ℓ = ( q − m − D ℓ ( i ). Lemma 3.3.
We have (a) | coker( B ) | = k . (b) The subgroups h (1 , i , h (0 , i , h (1 , i of coker( B ) have orders ke , kd , kf , respectively.Proof. (a) Define the matrix L = (cid:20) b b q − b b q − (cid:21) ∈ Z × , which can be regarded as a linear map L : Z → Z , whose cokernel iscoker( B ) = Z q − / h ( b , b ) , ( b , b ) i ∼ = Z / im( L ) . | Z / im( L ) | is invariant under elementary row or column operations (on L ). Therefore, we cansubstitute L by its Smith Normal form, and in particular | Z / im( L ) | is equal to the greatest common divisorof the determinants of the 2 × L , i.e. | coker( B ) | = | Z / im( L ) | = gcd(det( B ) , ( q − d, ( q − e ) = k. (b) It is enough to show that |h (1 , i| = k/e , since the other two are analogous. By definition, the order of(1 ,
0) is min { r ≥ r, ∈ im( L ) } = min (cid:8) r ≥ | coker( L ) | = | coker([ L | r ]) | (cid:9) . The greatest common divisor of the determinant of the 2 × L | r ] that do notappear in L is gcd( r ( q − , rb , rb ) = re . Therefore, |h (1 , i| = min { r ≥ k = gcd( k, re ) } = k/e . Proof of Theorem 2.2.
We prove (a), since the proofs of (b) and (c) are analogous. Note that the sets C ∗ ij for j = 0 , . . . , q − q − X j =0 |C ∗ ij | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − [ j =0 C ∗ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = |{ ( x, y ) ∈ ( F ∗ q ) : ρ i x a − a y a − a = 1 }| = ( q − − D d ( i )( q − . Therefore, q − X j =0 N ij = q − X j =0 (cid:0) |C ∗ ij | + D d ( i ) + D e ( j ) + D f ( i − j + w ) − ( q + 1) (cid:1) = ( q − − D d ( i )( q −
1) + D d ( i )( q −
1) + q − X j =0 D e ( j )+ q − X j =0 D f ( i − j + w ) − ( q + 1)( q − C ∗ i + b ,j + b → C ∗ ij given by ( x, y ) ( ρx, y ) is a bijection, so |C ∗ i + b ,j + b | = |C ∗ ij | . Moreover, D d ( i + b ) = D d ( i ), D e ( j + b ) = D e ( j ), and D f ( i − j + b − b + w ) = D f ( i − j + w )since d | b , e | b , and f | b − b by definition. This implies that N i + b ,j + b = N ij . The proof of N i + b ,j + b = N ij is analogous.Now we prove (e), X i,j |C ∗ ij | = X i,j ( N ij − D d ( i ) − D e ( j ) − D f ( i − j + w ) + q + 1) == X i,j N ij + X i,j D d ( i ) + X i,j D e ( j ) + X i,j D f ( i − j + w ) + ( q + 1) ( q − − X i,j N ij D d ( i ) − X i,j N ij D e ( j ) − X i,j N ij D f ( i − j + w ) + 2( q + 1) X i,j N ij + 2 X i,j D d ( i ) D e ( j ) + 2 X i,j D d ( i ) D f ( i − j + w ) + 2 X i,j D e ( j ) D f ( i − j + w ) − q + 1) X i,j D d ( i ) − q + 1) X i,j D e ( j ) − q + 1) X i,j D f ( i − j + w ) . By (a), (b), and (c) the sixth, seventh, eighth, and ninth terms vanish. The other terms can be calculatedby Lemma 3.1, thus X i,j |C ∗ ij | = X i,j N ij + ( q − ( q − q + 1 + d + e + f ) . (4)Note that |C ∗ ij | = |C ∗ ij × C ∗ ij | , C ∗ ij × C ∗ ij = (cid:26) ( x , y , x , y ) ∈ ( F ∗ q ) : ρ i x b y b + ρ j x b y b = 1 ρ i x b y b + ρ j x b y b = 1 (cid:27) . (cid:20) x b y b x b y b x b y b x b y b (cid:21) . The set C ∗ ij × C ∗ ij can be written as the disjoint union D ij ∪ E ij , where D ij = ( C ∗ ij × C ∗ ij ) ∩ { ( x , y , x , y ) ∈ ( F ∗ q ) : ∆ = 0 } and E ij = ( C ∗ ij × C ∗ ij ) ∩ { ( x , y , x , y ) ∈ ( F ∗ q ) : ∆ = 0 } .By Cramer’s rule, D ij = (cid:26) ( x , y , x , y ) ∈ ( F ∗ q ) : ∆ = 0 , ρ i = ( − x b y b + x b y b ) / ∆ ρ j = ( x b y b − x b y b ) / ∆ (cid:27) , which imply that the D ij are disjoint and their union is [ i,j D ij = (cid:26) ( x , y , x , y ) ∈ ( F ∗ q ) : ∆ = 0 , x b y b = x b y b x b y b = x b y b (cid:27) . Introducing the change of variables x = x /x and y = y /y , we get X ij |D ij | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)[ i,j D ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x, y ) ∈ ( F ∗ q ) : x b y b = 1 x b y b = 1 x b y b = x b y b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( q − (cid:18) ( q − − (cid:12)(cid:12)(cid:12) { x b y b = 1 } S ∪ { x b y b = 1 } S ∪ { x b − b y b − b = 1 } S (cid:12)(cid:12)(cid:12)(cid:19) . Note that S ∩ S = S ∩ S = S ∩ S = S ∩ S ∩ S , so | S ∪ S ∪ S | = | S | + | S | + | S | − | S ∩ S | . ByLemma 3.2, | S | = ( q − d , | S | = ( q − e , and | S | = ( q − f . Moreover, | S ∩ S | = | coker( B ) | = k . Alltogether, we get X i,j |D ij | = ( q − (cid:20) ( q − − ( q − d + e + f ) + 2 k (cid:21) . Observe that E ij = ( x , y , x , y ) ∈ ( F ∗ q ) : x b y b = x b y b x b y b = x b y b ρ i x b y b + ρ j x b y b = 1 , X i,j |E ij | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( i, j, x , y , x , y ) ∈ Z q − × ( F ∗ q ) : x b y b = x b y b x b y b = x b y b ρ i x b y b + ρ j x b y b = 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( x , y , x , y ) ∈ ( F ∗ q ) : x b y b = x b y b x b y b = x b y b (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = ( q − q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( x, y ) ∈ ( F ∗ q ) : x b y b = 1 x b y b = 1 (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = ( q − q − k. Now we have X i,j |C ∗ ij | = X ij |D ij | + X ij |E ij | = ( q − (cid:20) ( q − − ( q − d + e + f ) + qk (cid:21) . Finally, using Eq. (4), we get P i,j N ij = ( q − q ( k − d − e − f + 2). Proposition 3.4.
Let G be an abelian group and let g , g , g ∈ G such that G = h g , g i = h g , g i = h g , g i .Let K ≥ and let N : G → R be a function a N a := N ( a ) such that (a) X a ∈ g + h g i N a = 0 for all g ∈ G , (b) X a ∈ g + h g i N a = 0 for all g ∈ G , X a ∈ g + h g i N a = 0 for all g ∈ G , (d) X a ∈ G N a = K .Then | N g | ≤ s K (cid:18) | G | − n − n − n (cid:19) , for all g ∈ G , where n , n , and n are the orders of the elements g , g , and g , respectively.Proof. Let n = |h g i ∩ h g i| . The isomorphism G/ h g i ∼ = h g i / h g i ∩ h g i implies that n = n n | G | . Similarly, we define n and n .We study first the case when 1 + | G | = n + n + n . Assuming without loss of generality that n ≤ n ≤ n ,the previous equality implies that n <
3. The case n = 1 can only happen when g is the neutral elementof G , and then (a) reduces to N g = 0 for all g ∈ G . In the case n = 2, we have | G | = |h g , g i| ≤ n , hence + n + n = 1 + | G | ≥ n , and in particular n ≤
2. Therefore n = n = n = 2 and G is a group oforder | G | = 4. The elements g , g , g are pairwise distinct, since each pair of them generates the group, so G = { , g , g , g } ≃ Z ⊕ Z . Items (a), (b), (c) yield the following identities: N + N g = N g + N g = 0 N + N g = N g + N g = 0 N + N g = N g + N g = 0 , which imply N g = 0 for all g ∈ G , and the claim follows.Now we assume that 1 + | G | = n + n + n . We use Lagrange multipliers to get the desired upper boundfor N g . By the symmetry of the problem, we can restrict to the case N . Define the auxiliary function F = N + X ¯ g ∈ G/ h g i λ ¯ g X a ∈ g + h g i N a + X ¯ g ∈ G/ h g i µ ¯ g X a ∈ g + h g i N a + X ¯ g ∈ G/ h g i ε ¯ g X a ∈ g + h g i N a + γ − K + X a ∈ G N a ! with N a , λ ¯ g , µ ¯ g , ε ¯ g , and γ as independent variables. The critical points of F correspond with the localextrema of N subject to the restrictions stated in the theorem. Now, we calculate the partial derivatives of F with respect to each variable. With respect to λ ¯ g , µ ¯ g , ε ¯ g , and γ , we get the assumptions of the proposition.With respect to N a , we get ∂F∂N a = δ a, + λ ¯ a + µ ¯ a + ε ¯ a + 2 γN a = 0 (5)for all a ∈ G , where δ a, stands for the Kronecker delta.For any element g ∈ G , we have X a ∈ g + h g i ( δ a, + λ ¯ a + µ ¯ a + ε ¯ a + 2 γN a ) = χ h g i ( g ) + n λ ¯ g + X a ∈ g + h g i µ ¯ a + X a ∈ g + h g i ε ¯ a = χ h g i ( g ) + n λ ¯ g + n X ¯ a ∈ G/ h g i µ ¯ a + n X ¯ a ∈ G/ h g i ε ¯ a = 0 . Define λ = − n n P ¯ a ∈ G/ h g i µ ¯ a − n n P ¯ a ∈ G/ h g i ε ¯ a . The previous identity shows that λ ¯0 = λ − n and λ ¯ g = λ for all ¯ g = ¯0. Similarly, we define µ = − n n P ¯ a ∈ G/ h g i λ ¯ a − n n P ¯ a ∈ G/ h g i ε ¯ a , and then µ ¯0 = µ − n and6 ¯ g = µ for all ¯ g = ¯0. Analogously, we define ε = − n n P ¯ a ∈ G/ h g i λ ¯ a − n n P ¯ a ∈ G/ h g i µ ¯ a , and then ε ¯0 = ε − n and ε ¯ g = ε for all ¯ g = ¯0. By construction of ε , we have ε = − n n X ¯ a ∈ G/ h g i λ ¯ a − n n X ¯ a ∈ G/ h g i µ ¯ a = − n n (cid:18) | G | n λ − n (cid:19) − n n (cid:18) | G | n µ − n (cid:19) . Therefore λ + µ + ε = | G | , and Equation (5) can be rewritten as follows:2 γN a = − δ a, − | G | + χ h g i ( a ) n + χ h g i ( a ) n + χ h g i ( a ) n . (6)Squaring the previous equation and summing over all a ∈ G , we get4 γ X a ∈ G N a = 1 + 2 | G | − n − n − n = 0 . This allows us to get γ = 0, and together with Equation (6) for a = 0, concludes the proof. Proof of Corollary 2.3.
Consider G = Z q − / h ( b , b ) , ( b , b ) i = coker( B ). By Theorem 2.2(d), the map N : G → R such that ( i, j ) N ij is well defined. Let g = (1 , ∈ G , g = (0 , ∈ G , and g = (1 , ∈ G .With this notation, the hypotheses of Proposition 3.4 with K = kq ( k − d − e − f + 2) follow from Theorem 2.2and Eq. (3). By Lemma 3.3, we have n = k/e , n = k/d , and n = k/f . The only thing left to do is tosubstitute these values in Proposition 3.4 and a suitable rearrangement of the terms. We devote this section entirely to showing how to derive Thm. 1.1 as a consequence of Thm. 2.2. Weconsider the family of curves C ij = { ( x, y ) ∈ F p : ρ i x + ρ j y = 1 } for 0 ≤ i, j < p −
1. Removing the extra points on the lines x = 0, y = 0, z = 0, Gauss’ curve correspondsto { ( x, y ) ∈ ( F ∗ p ) : x + y = − } that has the same number of points as C ∗ . The number of points oneach of those lines is equal to the number of cubic roots of the unity in F p , which is equal to gcd(3 , p − M p = |C ∗ | + 3 gcd(3 , p − M p = p + 1 − D d (0) − D e (0) − D f ( w ) + N + 3 gcd(3 , p − , where d = e = f = gcd(3 , p −
1) and w = p − . Then M p = p + 1 + N .Case p d = e = f = 1, k = gcd( p − ,
9) = 1, then ˜ g = 0 and N ij = 0 for all0 ≤ i, j < p − N = 0 and M p = p + 1, as expected.Case p ≡ d = e = f = 3, k = gcd(3( p − ,
9) = 9 and ˜ g = ( k − d − e − f + 2) = 1.The cokernel of the matrix B = (cid:20) (cid:21) is Z ⊕ Z , so the numbers N ij reduce to only nine possibilities A = N N N N N N N N N depending on the class of ( i, j ) in coker( B ) by Thm. 2.2(d). Due to Thm. 2.2(a)(b)(c), each of the rows,columns and diagonals of the matrix A above adds up to zero. This proves that A = u v − u − vv − u − v u − u − v u v u, v ∈ Z . Moreover, by Eq. (3), the sum of the squares of the entries of A is 3( u + v +( u + v ) ) = 18 p ,so u + v + uv = 3 p. (7)Let ξ ∈ F p be a cubic root of the unity. Note that |C ∗ ij | is divisible by 9, since for each point ( x, y ) ∈ C ∗ ij ,its conjugates ( ξ r x, ξ s y ) are also in C ∗ ij for any 0 ≤ r, s <
3. By Eq. (2), u = N = |C ∗ | − ( p + 1) + D (0) + D (0) + D ( w ) = |C ∗ | − p + 8 ,v = N = |C ∗ | − ( p + 1) + D (0) + D (1) + D ( w −
1) = |C ∗ | − p + 2 . Therefore 2 v + v = 2 |C ∗ | + |C ∗ | − p −
4) is divisible by 9. Denoting ¯ v = v + u ∈ Z , Eq. (7) becomes u + 27¯ v = 4 p . Moreover, u = |C ∗ | − p + 8 ≡ u + 27¯ v = 4 p with u ≡ Z h − √− i is a UFD. The aim of this section is to calculate the genus of the projective closure C ij of the curve C ij in P ( F q ) inthe irreducible case. In order to do so, we use the standard formula that relates the genus of a curve withthe delta invariant δ P at each of its singularities, see formula (8) below. The delta invariants are computedusing the techniques shown in [1, Ch. 3 and 6]. The final formula obtained in Prop. 5.2 should be comparedterm by term to the definition of ˜ g given in Eq. (1). Lemma 5.1.
Let
C ⊆ P ( F q ) be a curve such that its local equation at P is given by αx r + βy s + γx u y v = 0 with αβ = 0 and r, s ≥ . If either γ = 0 or ( u, v ) is above the segment that joins ( r, and (0 , s ) , then δ P = 12 ( rs − r − s + gcd( r, s )) . If γ = 0 and ( u, v ) is below the segment, then δ P = 12 ( rv + su − r − s + gcd( u, s − v ) + gcd( v, r − u )) . The formula of the first case is valid even if r = 0 or s = 0 . Also, the formula of the second case is validwhen either r = u = 0 or s = v = 0 . In both situations, the point P does not belong to the curve and δ P = 0 .Proof. In the first case, the term γx u y v can be removed from the local equation without changing thetopology (since the point is above the Newton polygon). It is clear that the Milnor number at P is µ P =( r − s −
1) and that the number of local branches at P is r P = gcd( r, s ). Therefore 2 δ P = µ P + r P − rs − r − s + gcd( r, s ).In the other case, the local equation can be changed by αx r + βy s + γx u y v + αβγ x r − u y s − v , since the extra termis above the Newton polygon. Doing so, we get an expression that factorizes as ( αx u + αβγ y s − v )( x r − u + γα y v ).Applying the formula of the δ -invariant of a product, we get: δ P ( αx r + βy s + γx u y v ) = δ P ( αx u + αβγ y s − v ) + δ P ( x r − u + γα y v ) + i P ( αx u + αβγ y s − v , x r − u + γα y v ) , where i P denotes the intersection multiplicity at P . The values of δ P of each factor can be computed as inthe first case. Using Noether’s formula (see [2, p. 3568]), the intersection multiplicity is uv . We concludeby simply adding these three values. Proposition 5.2.
If the projective closure C ij of the curve C ij is irreducible in P ( F q ) , the genus of C ij is g ( C ij ) = 12 ( | det( B ) | − gcd( b , b ) − gcd( b , b ) − gcd( b − b , b − b ) + 2) . Proof.
By using the irreducibility of the curve, we can reduce the proof to the case a = a = 0 and a ≥ a . In this case, the genus can be computed using the following formula: g ( C ij ) = ( m − m − − X P δ P , (8)8here m is the degree of the curve, P ranges over all singular points of C ij , and δ P is the δ -invariant of C ij at P .Since we are assuming det( B ) = 0, the set of singular points is contained in { [1 : 0 : 0] , [0 : 1 : 0] , [0 : 0 : 1] } .We have to consider two cases: (1) m = a > a + a , (2) m = a + a ≥ a .Case (1): The projective closure of C ij is given by the homogeneous polynomial F ( x, y, z ) = ρ i x a + ρ j y a z a − a − x a y a z a − a − a . We can further assume without loss of generality that the point ( a , a ) is below the line that connectsthe points ( a ,
0) and (0 , a ), i.e. the Newton polygon of F has two edges. Otherwise, we would simplyexchange y by z and start over. Note that in the exceptional case when a = 0 ( a = 0), we should alsohave a = 0 ( a = 0), respectively. The singular points are [0 : 1 : 0] and [0 : 0 : 1], and the local equationsof C ij at those points are ρ i x a + ρ j z a − a − x a z a − a − a and ρ i x a + ρ j y a − x a y a , respectively.By Lemma 5.1, we have δ [0:1:0] = 12 ( a ( a − a ) − a − ( a − a ) + gcd( a , a − a )) ,δ [0:0:1] = 12 ( a a + a a − a − a + gcd( a , a − a ) + gcd( a , a − a )) . Finally, using (8), we get the desired formula.Case (2): The projective closure of C ij is given by the homogeneous polynomial F ( x, y, z ) = ρ i x a z a + a − a + ρ j y a z a + a − a − x a y a . The local equations of C ij at [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1] are ρ i z a + a − a + ρ j y a z a + a − a − y a , ρ i x a z a + a − a + ρ j z a + a − a − x a , and ρ i x a + ρ j y a − x a y a , respectively. By Lemma 5.1, δ [1:0:0] = 12 ( a ( a + a − a ) − a − ( a + a − a ) + gcd( a , a − a )) ,δ [0:1:0] = 12 ( a ( a + a − a ) − a − ( a + a − a ) + gcd( a , a − a )) ,δ [0:0:1] = 12 ( a a − a − a + gcd( a , a )) . The conclusion follows from formula (8).
We study some particular cases of Theorem 2.2 and Corollary 2.3 which have special significance by them-selves.
Example . The curve C ij = { ( x, y ) ∈ ( F ∗ q ) : ρ i x a + ρ j y a = 1 } has d = gcd( a , q − e = gcd( a , q − f = gcd( d, e ), and w = ( q − / q odd, and w = 0 otherwise. Moreover,coker( B ) = Z q − / h ( a , , (0 , a ) i ∼ = Z d ⊕ Z e , hence k = | coker( B ) | = de . By Theorem 2.2(d), we have N i + d,j = N ij = N i,j + e , so the matrix [ N ij ] ≤ i,j 1. Similarly, the sum (b) can be takenfrom i = 0 to i = d − Example . We consider the subcase of Example 6.1 when a is odd, a = 2, and q is odd. The constants d , e , f , k , w reduce to d = gcd( a , q − e = 2, f = 1, k = 2 d , w = ( q − / 2, and the upper-left blockis of size d × 2. Since the second column of this block is the additive inverse of the first one, we have, asmultisets: { N ij : 0 ≤ i, j < q − } = ( q − d · {± α , ± α , . . . , ± α d − } α i = N i . Moreover, α + α + · · · + α d − = 0 ,α + α + · · · + α d − = d ( d − q. The vector ( α , . . . , α d − ) is in the intersection of a sphere and a hyperplane in R d , i.e. the vector( α , . . . , α d − ) belongs to a conic in R d − . Of course, when d = 1, the sphere reduces to a point. Example . Now, we consider the curves C ij = { ( x, y ) ∈ ( F ∗ q ) : ρ i x + ρ j y = 1 } , which is a particularcase of the previous example. Clearly, when q d = 1 and all the N ij are zero. Forthis reason, we only consider q ≡ d = 3: { N ij : 0 ≤ i, j < q − } = ( q − · {± α , ± α , ± α } , where α + α + α = 0 and α + α + α = 6 q . • If q = p n for some p ≡ α = p n β , α = p n β , and α = p n β , for some β , β , β ∈ Z such that β + β + β = 0 and β + β + β = 6. This implies that, as multisets: { N ij : 0 ≤ i, j < q − } = ( q − · {± p n , ± p n , ∓ p n } . In particular, N ij = 0 for all i, j and the upper bound of Theorem 2.2 is sharp for this family. • In constrast, for p ≡ q = p = 997, wehave α = 10, α = 49, α = − 59, but the integer part of the upper bound is ⌊ ( k − d − e − f +2) √ q ⌋ = 63.Note, however, that max x : x, y, z ∈ Z x + y + z = 0 x + y + z = 6 q = 59 , (9)so one may think that the largest N ij can be obtained always by solving optimization problem inProp. 3.4 for a function f : G → Z . • In the case q = 7 , we have { N ij : 0 ≤ i, j < } = 384 · {∓ , ∓ , ± } , so the largest N ij is 13.However, the integer optimization problem (9) gives 14. This highlights the fact that the relationsgiven in Thm 2.2 are not always enough to characterize the maximum N ij . Example . When a = 2, q is odd, and a is even, the situation is similar, but f = 2, and item (c) ofThm 2.2, gives the additional relation α − α + · · · + α d − − α d − = 0. All together this gives α + α + · · · + α d − = 0 ,α + α + · · · + α d − = 0 ,α + α + · · · + α d − = d ( d − q. Example . The curve C ij = { ( x, y ) ∈ ( F ∗ q ) : ρ i x + ρ j y = x } with odd q , has d = 2, e = f = 1, k = gcd( q − , w = ( q − / 2. When q ≡ k = 2, so k − d − e − f + 2 = 0, and inparticular N ij = 0 for all i, j . The other case, i.e. q ≡ k = 4, andcoker( B ) = Z q − / h (2 , , ( − , i = { (0 , , (1 , , (0 , , (1 , } . By Thm 2.2(d), N i +2 ,j = N ij = N i − ,j +2 , so as multisets { N ij : 0 ≤ i, j < q − } = ( q − · {± α, ± β } , where α = N , β = N , and α + β = 4 q . If, we also have p ≡ q = p n and the multisetis ( q − · {± p n , } . 10 eferences [1] E. Casas-Alvero: Singularities of Plane Curves. London Mathematical Society Lecture Note Series 276.Cambridge University Press, 2000.[2] J.I. Cogolludo, J. Martin-Morales, J. Ortigas-Galindo: Local Invariants on Quotient Singularities anda Genus Formula for Weighted Plane Curves. International Mathematics Research Notices, Vol. 2014,No. 13, pp. 3559–3581.[3] P. Deligne: La conjecture de Weil. I. (French) Inst. Hautes ´Etudes Sci. Publ. Math. No. 43, pp. 273–307,1974.[4] B. Dwork: On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82, pp. 631–648, 1960.[5] C.F. Gauss: Disquisitiones Arithmeticae (translated by A. Clarke). Yale University Press, 1966.[6] A. Grothendieck: Formule de Lefschetz et rationalit´e des fonctions L. S´eminaire Bourbaki, Vol. 9,Exp. No. 279, pp. 41–55, 1964–1966.[7] L. Hua, H.S. Vandiver: On the number of solutions of some trinomial equations in a finite field.Proceedings of the National Academy of Sciences of the USA, Vol. 35, No. 8, pp. 477–481, 1949.[8] J. Silverman: The arithmetic of elliptic curves. Graduate texts in mathematics 106. Springer-Verlag,New York, 1986.[9] J. Silverman, J. Tate: Rational points on elliptic curves. Springer-Verlag, New York, 1992.[10] A. Weil: Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc. 55, pp. 497–508,1949. M. Avendano, Centro Universitario de la Defensa, Academia General Militar, Ctra. de Huesca s/n., 50090,Zaragoza, Spain E-mail address : [email protected] J. Mart´ın-Morales, Centro Universitario de la Defensa, Academia General Militar, Ctra. de Huesca s/n., 50090,Zaragoza, Spain E-mail address : [email protected]@unizar.es