Number of points in the intersection of two quadrics defined over finite fields

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1 Number of points in the intersection of two quadrics defined over finite fields Frédéric A. B. EDOUKOU Nanyang Technological University SPMS-MAS Singapore Carleton Finite Fields Workshop July 20-23, 2010 Carleton University Ottawa, (Ontario), Canada 1

2 Contents Notations I- Some results on intersection of quadrics II-Intersection of two quadrics in P 3 (F q ) III-Intersection of two quadrics in P 4 (F q ) IV-Applications to coding theory V-Generalization to quadrics in P n (F q ) VI-Conclusion References 2

3 Notations F q : finite field with q elements, (q = p a ). V = A m+1 aff. sp. of dim. m + 1 on F q. P m (F q ): proj. space of dim. m. #P m (F q ) = q m + q m q + 1 π m = q m + q m q + 1 F h (V, F q ): forms of degree h on V with coefficients in F q. 3

4 I-Some results on intersection of two quadrics in P n (F q ). In 1975, W. M. Schmidt Q 1 Q 2 2(4q n 2 + 4π n 3 ) + 7 q 1 In 1992, Y. aubry Q 1 Q 2 2(4q n 2 + π n 3 ) + 1 q 1 In 1999, D. B. Leep et L. M. Schueller Suppose: w(q 1, Q 2 ) = n + 1 If n and even, then: Q 1 Q 2 2q n 2 + π n 3 + 2q n 1 2 3q n 3 2 If n and odd, then Q 1 Q 2 2q n 2 + π n 3 + q n 2 In 2006, Lemma Let 1 l n 1 and w(q 1, Q 2 ) = n l + 1. If Q 1 Q 2 E m where E P n l (F q ), then Q 1 Q 2 mq l + π l 1 This bound is the best possible as soon as m is optimal for E. 4

5 II-Intersection of two quadrics in P 3 (F q ) X : F (x 0, x 1, x 2, x 3 ) = 0 Table 1: Quadrics in PG(3,q). r(q) Description Q g(q) 1 repeated plane π 2 2 Π 2 P 0 2 pair of distinct planes 2q 2 + π 1 2 Π 2 H 1 2 line π 1 1 Π 1 E 1 3 quadric cone π 2 1 Π 0 P 2 4 hyperbolic quadric π 2 + q 1 H 3 (R, R ) 4 elliptic quadric π 2 q 0 E 3 Some values de #X Z(f) (F q ) C(q) = 4q + 1, C 2 (q) = 3q + 1, C 3 (q) = 3q H(q) = 4q, H 2 (q) = 3q + 1, H 3 (q) = 3q E(q) = 2(q + 1), E 2 (q) = 2q + 1, E 3 (q) = 2q 5

6 III-Intersection of two quadrics in P 4 (F q ) Table 2: Quadrics in P 4 (F q ). r(q) Description Q g(q) 1 repeated hyperplane π 3 Π 3 P pair of hyperplanes 2q 3 + π 2 Π 2 H plane π 2 Π 2 E cone π 3 Π 1 P cone π 3 + q 2 Π 0 H 3 (R, R ) 2 4 cone π 3 q 2 Π 0 E parabolic quadric π 3 P 4 1 6

7 Section of X: g(q)=2 Table 3: Plane quadric curves r(q ) Description Q g(q ) 1 repeated line Π 1 P 0 q pair of lines Π 0 H 1 2q point Π 0 E parabolic P 2 q #X Z(f) (F q ) 2q 2 + 3q + 1 Section of X: g(q)=3 a. Q is a repeated hyperplane Theorem [Primrose, 1951] Let H P 4 (F q ) be an hyperplane #X H (F q ) = π 2 + q, π 2 q if H n.-tan. to X, π 2 if H is tan. to X. 7

8 b. Q is a pair of hyperplanes: Q = H 1 H 2 ˆX 1 = H 1 X, ˆX 2 = H 2 X et P = H 1 H 2 Q X = H 1 X + H 2 X P X. (1) P X = P ˆX 1 = P ˆX 2. (2) Theorem [Swinnerton-Dyer, 1964 ] Let X be a degenerate quadric variety of rank r < n+1 in P n (F q ) and Π r 1 a linear projective space of dimension r 1 disjoint from the singular space Π n r of X. Then Π r 1 X is a nondegenerate quadric variety in Π r 1. Theorem [Wolfmann, 1975 ] Let X P n (F q ) be a non-degenerate quadric variety. A tangent hyperplane meets X at a denegerate quadric of the same type as X. 8

9 b.1 Two tangent hyperplanes to Q b.2 One tangent and one n-tang. to Q b.3 Two tangent hyperplanes to Q Proposition If Q is a pair of hyperplanes in P 4 (F q ) and X the non-degenerate quadric variety in P 4 (F q ), then #X Z(Q) (F q ) = 2q 2 + 3q + 1, 2q 2 + 2q + 1 #X Z(Q) (F q ) = 2q 2 + q + 1, 2q 2 + 1, #X Z(Q) (F q ) = 2q 2 q + 1 9

10 Section of X : g(q)=1 a. X Q contains no line #X Z(f) (F q ) 2(q 2 + 1) b. X Q contains some lines b.1. Q est degenerate #X Z(f) (F q ) 2q 2 + 2q + 1 b.2. Q is non-degenerate Table 4: Intersection of ˆQ i ˆX i in P 3 (F q ) Type ˆQ i ˆX i 1 (hyperbolic quadric) (quadric cone) 2 (quadric cone) (quadric cone) 3 (hyperbolic quadric) (hyperbolic quadric) 10

11 Table 5: Number of points and lines in ˆQ i ˆX i Types 4 lines 3 lines 2 lines 1 line 1 3q 2q q+1 3q+1 2q + 1 2q q 3q+1 3q + 1 2(q + 1) A) X Q contains exactly one line #X Z(f) (F q ) q 2 + 3q + 2 B) X Q contains at least two lines: B-1) X Q contains only skew lines #X Z(f) (F q ) q 2 + 3q + 2 B-2) X Q contains at least two secant lines: ( )It exists H 1 and H 2 such that ˆX i = ˆQ i #X Z(f) (F q ) 2q 2 + 2q + 1 ( ) It exists H 1 such that ˆX 1 = ˆQ 1 #X Z(f) (F q ) q 2 + 6q + 2 ( ) For i = 1,..., q + 1 ˆX i ˆQ i #X Z(f) (F q ) 2q 2 + 3q

12 Some values of #X Z(f) (F q ) Theorem If X is a non-degenerate quadric in P 4 (F q ) and Q a quadric of P 4 (F q ) such that X λq, then #X Z(Q) (F q ) = 2q 2 + 3q + 1, 2q 2 + 2q + 1 #X Z(Q) (F q ) = 2q 2 + q + 1, 2q 2 + 1, #X Z(Q) (F q ) = 2q 2 q + 1 Theorem Let X be a quadric in P 4 (F q ) and Q another quadric in P 4 (F q ). If X is : non-degenerate, then #X Z(Q) (F q ) 2q 2 + 3q + 1. degenerate with r(x) = 3, then #X Z(Q) (F q ) 4q 2 + 3q + 1. degen. with r(x) = 4 and g(x) = 2 then, #X Z(Q) (F q ) 4q These bounds are the best possible. 12

13 IV- Applications to Coding Theory Let X P m (F q ) and N = #X(F q ) c : F h (V, F q ) F N q f c(f) = (f(p 1 ),..., f(p N )) C h (X) = Imc definition Let c(f) be a codeword cw(f) = #{P X f(p ) = 0} w(c(f)) = #X(F q ) cw(f) distc h (X) = min f F h {w(c(f))} Proposition The parameters of C h (X): lenght C h (X) = #X(F q ), dim C h (X) = dim F h dim ker c, distc h (X) = #X(F q ) max f F h #X Z(f) (F q ) 13

14 Weights Distribution of C 2 (H 3 ) w 1 = q 2 2q + 1. The codewords << w 1 >>: union of 2 tan planes and l bisecant hyperbolic quadric containing ll and = lines of X. w 2 = q 2 q. The codewords << w 2 >>: hyperbolic quadric containing exactly two lines in distinct regulus and the q other lines of one regulus are bisecants of X. union of two tangent planes of X and the line of intersection is contained in X. union of two planes one is tan., the second is non-tan. to X and the line of intersection intersecting X at a single point. w 3 = q 2 q + 1. The codewords << w 3 >>: union of two planes one tan., the second non-tan. to X and the line of intersection intersecting X at two points. 14

15 Weights Distribution of C 2 (E 3 ) w 1 = q 2 2q 1 The codewords << w 1 >>: union of two planes non-tan and l disjoint to X. hyperbolic quadrics with all lines of one regulus are bisecants. degenerate quadrics of rank 3 (i.e. q +1 lines) with the vertex no contained in X et and all the q + 1 lines are bisecants. w 2 = q 2 2q The codewords << w 1 >>: quadrics which are union of two non-tan. planes to X and the line of intersection intersecting X at two points. w 3 = q 2 2q

16 Table 6: The first 5 weights of C 2 (X). Numb Q P X w i 1 2 n-tan H n-sin. conic q 3 q 2 2q 2 2 n-tan sin. cve (r=2) q 3 q 2 q 3 1t+1n-tan Π 0 H 1 q 3 q 2 4 1t+1n-tan sin. cve (r=2) 2tan Π 0 H 1 q 3 q 2 + q 5 2 n-tan E n-sin. conic q 3 q 2 + 2q Tuesday 02/27/2007 (Seminar of GRIM, Toulon) Theorem [Ax, 1964] Let r polynomials f i (x 1,..., x n ) and deg (f i ) = d i on F q then: if n > b r i=1 d i q b #Z(f 1,..., f n ). 16

17 V-Generalization to quadrics in P n (F q ) Theorem[E., Hallez, Rodier, Storme] Let X be a quadric in P n (F q ) with n 5, If X Q q n 2 +3q n 3 +3q n 4 +2q n q+1, then there exist a quadric consisting of two hyperplanes. 17

18 5.1 X is a non-degenerate quadric in P 2l+1 (F q ) X Q 2q n 2 + π n 3 + 2q n 1 2 q n 3 2 Weights Distribution of C 2 (H 2l+1 ) distc h (X): D. Leep, in 1999, FFA (7). distc h (X) q 2l q 2l 1 q l + q l 1 Table 6: The first 6 weights of C 2 (X). Numb Q Π 2l 1 X w i 1 2 tan. H 2l 1 q 2l q 2l 1 q l + q l 1 2 1t+1n-tan Π 0 P 2l 2 2tan Π 1 H 2l 3 q 2l q 2l 1 3 1t+1n-tan H 2l 1 q 2l q 2l 1 + q l n-tan. E 2l 1 q 2l q 2l 1 + q l q l n-tan Π 0 P 2l 2 q 2l q 2l 1 + q l 6 2 n-tan H 2l 1 q 2l q 2l 1 + q l + q l 1 18

19 Weights distribution of C 2 (E 2l+1 ) distc h (X): D. Leep, in 1999, FFA (7). distc h (X) q 2l q 2l 1 3q l + 3q l 1 Table 7: The first 7 weights of C 2 (E 2l+1 ). Numb Q Π 2l 1 X w i 1 2 n-tan. E 2l 1 q 2l q 2l 1 q l q l n-tan Π 0 P 2l 2 q 2l q 2l 1 q l 3 1t+1n-tan H 2l 1 q 2l q 2l 1 q l + q l n-tan. E 2l 1 q 2l q 2l 1 q l 1 5 1t+1n-tan Π 0 P 2l 2 2tan Π 1 E 2l 3 q 2l q 2l tan E 2l 1 q 2l q 2l 1 + q l q l tan Π 0 P 2l 2 q 2l q 2l 1 + q l Theorem [E. Hanja, Rodier Storme] Let X be a non-degenerate quadric in P 2l+1 (F q ) with l N. Then all the weights of the code C 2 (X) defined on X are divisible by q l 1. 19

20 5.2 X is a non-degenerate quadric in P 2l+2 (F q ) distc h (X): D. Leep, in 1999, FFA (7). distc h (X) q 2l+1 q 2l q l+1 Table 8: the first 5 weights of C 2 (P 2l+2 ). Numb Q Π 2l X w i 1 2 n-tan. H P 2l q 2l+1 q 2l 2q l 2 n-tan Π 0 H 2l 1 2 1tan+1n-tan P 2l q 2l+1 q 2l q l 2 tan Π 0 E 2l 1 2 n-tan P 2l 3 1tan+1n-tan Π 0 H 2l 1 q 2l+1 q 2l 1tan+1n-tan. Π 0 E 2l 1 2 tan Π 1 P 2l 2 2 n-tan. H Π 0 E 2l 1 4 1tan+1n-tan P 2l q 2l+1 q 2l + q l 2 tan Π 0 H 2l n-tan E P 2l q 2l+1 q 2l + 2q l Theorem [E., Hanja, Rodier, Storme] Let X be a non-degenerate quadric in P 2l+2 (F q ) and l N. Then all the weights of the code C 2 (X) defined on X are divisible by q l. 20

21 VII-Conclusion Theorem [E., San, Xing] Let Q 1 and Q 2 be two quadrics in P n (F q ) with no common d hyperplane. Then: Q 1 Q 2 4q n 2 + π n 3 Conjecture [E., San, Xing] Let X P n (F q ) be an arbitrary algebraic set of dimension s and degree d. Then the number of X is such that: #X(F q ) dq s + π s

22 References E. J. F. Primrose, Quadrics in finite geometries. Proc. Camb. Phil. Soc., (1951), F. A. B. Edoukou, Codes defined by forms of degree 2 on quadric surfaces. IEEE, Trans. Inf. Theory, Vo. 54, (2), (2008), F. A. B. Edoukou,Codes defined by forms of degree 2 on quadric varieties in P 4 (F q ). Contemp. Mathematics 487, (2009), F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, A study of intersections of quadrics having applications on the small weight codewords of the functional codes. C 2 (Q), Q a non-singular quadric. J.P.A.A, 214, (2010), F. A. B. Edoukou, S. Ling and C. Xing, Intersection of two quadrics with no common on hyperplanes. Submitted. 22

23 Thank you for your attention 23