Number of points in the intersection of two quadrics defined over finite fields Frédéric A. B. EDOUKOU e.mail:abfedoukou@ntu.edu.sg Nanyang Technological University SPMS-MAS Singapore Carleton Finite Fields Workshop July 20-23, 2010 Carleton University Ottawa, (Ontario), Canada 1
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
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 1 +... + q + 1 π m = q m + q m 1 +... + q + 1 F h (V, F q ): forms of degree h on V with coefficients in F q. 3
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 + 1 4 and even, then: Q 1 Q 2 2q n 2 + π n 3 + 2q n 1 2 3q n 3 2 If n + 1 5 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
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
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 0 3 2 pair of hyperplanes 2q 3 + π 2 Π 2 H 1 3 2 plane π 2 Π 2 E 1 2 3 cone π 3 Π 1 P 2 2 4 cone π 3 + q 2 Π 0 H 3 (R, R ) 2 4 cone π 3 q 2 Π 0 E 3 1 5 parabolic quadric π 3 P 4 1 6
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 + 1 1 2 pair of lines Π 0 H 1 2q + 1 1 2 point Π 0 E 1 1 0 3 parabolic P 2 q + 1 0 #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
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
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
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
Table 5: Number of points and lines in ˆQ i ˆX i Types 4 lines 3 lines 2 lines 1 line 1 3q 2q + 1 2 4q+1 3q+1 2q + 1 2q + 1 3 4q 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 + 1 11
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 2 + 1. These bounds are the best possible. 12
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
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
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 + 1 15
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
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 5 +...+2q+1, then there exist a quadric consisting of two hyperplanes. 17
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 1 4 2 n-tan. E 2l 1 q 2l q 2l 1 + q l q l 1 5 2 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
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 1 2 2 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 1 4 2 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 1 6 2 tan E 2l 1 q 2l q 2l 1 + q l q l 1 7 2 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
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 1 5 2 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
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 1.. 21
References E. J. F. Primrose, Quadrics in finite geometries. Proc. Camb. Phil. Soc., (1951), 299-304. F. A. B. Edoukou, Codes defined by forms of degree 2 on quadric surfaces. IEEE, Trans. Inf. Theory, Vo. 54, (2), (2008), 860-864. F. A. B. Edoukou,Codes defined by forms of degree 2 on quadric varieties in P 4 (F q ). Contemp. Mathematics 487, (2009), 21-32. 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), 1729-1939. F. A. B. Edoukou, S. Ling and C. Xing, Intersection of two quadrics with no common on hyperplanes. Submitted. 22
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