How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
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1 How Are Irreducible and Primitive Polynomials Distributed over Finite Fields? Gary L. Mullen Penn State University July 21, 2010 How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
2 Let q = p m be a prime power Let F q = F p m denote the finite field of order q How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
3 Primitive Polynomials f F q [x] of deg. n is primitive if every root of f is a prim. ele. in F q n Recall that a prim. ele. in F q n generates the mul. group F q n How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
4 Cohen (Disc. Math., 90) Let n 2 and let a F q (with a 0 if n = 2 or if n = 3 and q = 4). Then there exists prim. deg. n over F q with trace a. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
5 Cohen (Disc. Math., 90) Let n 2 and let a F q (with a 0 if n = 2 or if n = 3 and q = 4). Then there exists prim. deg. n over F q with trace a. Conjecture Hansen/M (Math. Comp, 92) Conj A: For n 2, 0 j < n and given a F q, there is prim. x n + + ax j + except when (P1) q arb., j = 0, a ( 1) n α, α prim. in F q (P2) q arb., n = 2, j = 1, a = 0 (P3) q = 4, n = 3, j = 2, a = 0 (P4) q = 4, n = 3, j = 1, a = 0 (P5) q = 2, n = 4, j = 2, a = 1 How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
6 Many papers by various people culminating in Cohen (FFA 06) Conj. A is true for deg. n 9. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
7 Many papers by various people culminating in Cohen (FFA 06) Conj. A is true for deg. n 9. Cohen/Presern (Lond. Math. Soc. Lect. Note Ser. 07) Conj. A is true!! How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
8 Other Related Results Han (Math. Comp., 96) For q odd and n 7, prim. deg. n with coeffs. of x n 1 and x n 2 given in advance. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
9 Other Related Results Han (Math. Comp., 96) For q odd and n 7, prim. deg. n with coeffs. of x n 1 and x n 2 given in advance. Han (FFA 97) For q even and n 7, prim. deg. n with coeff. x n 1 and x n 2 given in advance. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
10 Other Related Results Han (Math. Comp., 96) For q odd and n 7, prim. deg. n with coeffs. of x n 1 and x n 2 given in advance. Han (FFA 97) For q even and n 7, prim. deg. n with coeff. x n 1 and x n 2 given in advance. Cohen/Mills (FFA 03) For q odd and n = 5, 6 prim. deg. n with coeff. x n 1 and x n 2 given in advance. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
11 Other Related Results Han (Math. Comp., 96) For q odd and n 7, prim. deg. n with coeffs. of x n 1 and x n 2 given in advance. Han (FFA 97) For q even and n 7, prim. deg. n with coeff. x n 1 and x n 2 given in advance. Cohen/Mills (FFA 03) For q odd and n = 5, 6 prim. deg. n with coeff. x n 1 and x n 2 given in advance. Shuqin/Han (FFA 04) For n 8 prim. deg. n with highest three coeff. given in advance. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
12 Problem Find formulas, or good estimates, for the # of prim. deg. n over F q with given trace, (or even more coeff). specified in advance. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
13 Problem Find formulas, or good estimates, for the # of prim. deg. n over F q with given trace, (or even more coeff). specified in advance. Chang/Chou/Shiue (FFA 05) Enum. results. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
14 Primitive Normal Polynomials For α F q n, if A = {α, α q,..., α qn 1 } is a basis over F q, A is normal basis. If α = Fq n, A is prim. nor. basis. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
15 Carlitz (Trans. 52) Prim. nor. basis over F p for suff. large p. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
16 Carlitz (Trans. 52) Prim. nor. basis over F p for suff. large p. Davenport (J. Lond. M. S. 68) Prim. nor. basis over F p for any prime p. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
17 Carlitz (Trans. 52) Prim. nor. basis over F p for suff. large p. Davenport (J. Lond. M. S. 68) Prim. nor. basis over F p for any prime p. Lenstra-Schoof (Math. Comp., 87) F q n has prim. nor. basis over F q. How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
18 Carlitz (Trans. 52) Prim. nor. basis over F p for suff. large p. Davenport (J. Lond. M. S. 68) Prim. nor. basis over F p for any prime p. Lenstra-Schoof (Math. Comp., 87) F q n has prim. nor. basis over F q. Cohen/Huczynska (J. London. M. Soc. 03) Prim. nor. basis without a computer! How Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
19 φ(q n 1) is # of prim. elem. in F q n Φ q (x n 1) is # of nor. basis elem. in F q n. Problem Find a formula for the number P N q (n) of prim. nor. elem. in F q n. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
20 Conjecture Morgan/Mul. (Math. Comp., 94) For n 2 and a F q, there is a prim. nor. poly. deg. n over F q with trace a. True for: q = 2 any n n = 2 any q (q 1) n n 6, q 97 How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
21 Cohen/Hac. (AAECC, 99) Morgan/M Conj. true How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
22 Cohen/Hac. (AAECC, 99) Morgan/M Conj. true Conjecture Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n 2, prim. nor. poly. deg. n over F p with at most 5 nonzero coeffs. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
23 Cohen/Hac. (AAECC, 99) Morgan/M Conj. true Conjecture Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n 2, prim. nor. poly. deg. n over F p with at most 5 nonzero coeffs. Cohen (Pub. Math. Deb. 00) Prim. nor. of deg. n 5 with given norm and trace. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
24 Cohen/Hac. (AAECC, 99) Morgan/M Conj. true Conjecture Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n 2, prim. nor. poly. deg. n over F p with at most 5 nonzero coeffs. Cohen (Pub. Math. Deb. 00) Prim. nor. of deg. n 5 with given norm and trace. Cohen/Huczynska (Acta Arith, 03) Prim. nor. quartics with given norm and trace How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
25 Cohen/Hac. (AAECC, 99) Morgan/M Conj. true Conjecture Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n 2, prim. nor. poly. deg. n over F p with at most 5 nonzero coeffs. Cohen (Pub. Math. Deb. 00) Prim. nor. of deg. n 5 with given norm and trace. Cohen/Huczynska (Acta Arith, 03) Prim. nor. quartics with given norm and trace Huczynska/Cohen (Trans. A.M.S. 03) Prim. nor. cubics with given norm and trace How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
26 Fan/Wang (FFA 09) If n 15, prim. nor. with any coeff. specified in advance How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
27 Completely Normal Bases F q F q d F q n α F q n nor. basis over F q and F q d? α F q n nor. basis over F q d for all d n? Blessenohl/Johnsen (J. Alg., 86) F q n has a com. nor. basis. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
28 Conjecture Morgan/M (Util. Math. 96) For each n 2 there is a com. nor. prim. poly. deg. n over F q. True for: n = 4 q n 2 31, q 97 Shparlinski/Mul. (Finite Fields Appl., CUP, 96) For q Cnlog n, com. nor. prim. basis of F q n over F q. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
29 Conjecture Morgan/M (Util. Math. 96) For each n 2 there is a com. nor. prim. poly. deg. n over F q. True for: n = 4 q n 2 31, q 97 Shparlinski/Mul. (Finite Fields Appl., CUP, 96) For q Cnlog n, com. nor. prim. basis of F q n over F q. Problem Find formula for the number CN q (n) of com. nor. bases of F q n over F q. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
30 Irreducibles Conjecture Hansen/M (Math. Comp. 92) Conj B: For n 2, 0 j < n and given a F q there is irr. x n + + ax j + except (I1) q arb. j = a = 0 (I2) q = 2 m, n = 2, j = 1, a = 0 How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
31 Irreducibles Conjecture Hansen/M (Math. Comp. 92) Conj B: For n 2, 0 j < n and given a F q there is irr. x n + + ax j + except (I1) q arb. j = a = 0 (I2) q = 2 m, n = 2, j = 1, a = 0 Wan (Math. Comp., 97) If either q > 19 or n 36, Conj. B is true. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
32 Irreducibles Conjecture Hansen/M (Math. Comp. 92) Conj B: For n 2, 0 j < n and given a F q there is irr. x n + + ax j + except (I1) q arb. j = a = 0 (I2) q = 2 m, n = 2, j = 1, a = 0 Wan (Math. Comp., 97) If either q > 19 or n 36, Conj. B is true. Ham/Mul. (Math. Comp. 97) Conj. B is true. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
33 Hsu (J. Numb. Thy., 96) (i) If f has even deg. n and q n/2 + 1, irr. P of deg. n with deg. (P f) at most n/2. (ii) If f has odd deg. n and q ((n + 3)/2) 2, irr. P of deg. n with deg. (P f) at most (n 1)/2. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
34 N q (n) = # monic irr. deg. n Exact Formulas = 1 µ(d)q n/d n d n # monic irr. deg. n trace 1 1 2n d n,dodd µ(d)2 n/d How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
35 Fix 1 j n, β F 2 n T j (β) = β 2i1 β 2i2 β 2i j 0 i 1 <i 2 < <i j n T j : F 2 n F 2 T 1 (β) = β + β 2 + β β 2n 1 How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
36 F (n, t 1,..., t r ) = #β F 2 n with T j (β) = t j, j = 1,..., r P (n, t 1,..., t r ) = # irr. deg n with coeff. x n j = t j, j = 1,..., r P (n, 0, 0, 1) = #x n + 0x n 1 + 0x n 2 + 1x n np (n, 0, 0, 1) = µ(d)f (n/d, 0, 0, 1) d n,dodd How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
37 Cattell/Miers/Ruskey/Serra/Sawada (JCMCC 03) F (n, t 1, t 2 ) = 2 n 2 + G(n, t 1, t 2 ), m(mod4) G(n, t 1, t 2 ) = 0 2 m 1 2 m m 1 2 m m 1 2 m m 1 2 m 1 Kuz min (Sov. Math. Dokl. 91) How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
38 Yucas/M (Disc. Math. 04) For n even, F (n, t 1, t 2, t 3 ) = 2 n 3 + G(n, t 1, t 2, t 3 ), m(mod12) m 2 m 2 2 m m 2 2 m 2 2 m 2 1or5 2 m 2 2 m 2 2 m 2 2 m 2 2or m m m 2 2 m 2 2 m 2 2 m 2 4or8 2 m m 1 2 m 6 2 m m 2 2 m 2 2 m 1 2 m 2 2 m 2 7or11 2 m 2 2 m 2 2 m 2 2 m m 2 2 m 2 2 m 2 2 m 2 How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
39 m(mod12) or5 2 m 2 2 m 2 2 m m 2 2 m 2 2or10 2 m 1 2 m 1 2 m 1 2 m m m 1 2 m 4or m 1 2 m 1 2 m 1 2 m 1 7or11 2 m 2 2 m m 2 2 m 2 2 m m 2 m m 1 How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
40 Conjecture Yucas/Mul If n = 2m, F (n, t 1,..., t r ) = 2 n r + a m s+1 2 m s a m 2 m 1 s m, a i = 1, 0, 1 How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
41 Fitzgerald/Yucas (FFA 03) Formula for F (n, t 1, t 2, t 3 ) for odd n. non-deg., alt., sym., bil., quad. forms How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
42 Problem Extend to more than three coeff. over F 2. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
43 Problem Extend to more than three coeff. over F 2. Problem Extend two coeff. case to F q for q 2. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
44 Problem Extend to more than three coeff. over F 2. Problem Extend two coeff. case to F q for q 2. Problem Extend to j coeff. over F q. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
45 Several Variables Corteel/Savage/Wilf/Zeilberger, (JCT,A 98) The # of pairs of polys. f(x) and g(x) of deg. m over F 2 with (f, g) = 1 is the same as the # of pairs of polys. of deg. m with (f, g) 1. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
46 Several Variables Corteel/Savage/Wilf/Zeilberger, (JCT,A 98) The # of pairs of polys. f(x) and g(x) of deg. m over F 2 with (f, g) = 1 is the same as the # of pairs of polys. of deg. m with (f, g) 1. Benjamin/Bennett, (Math. Mag. 07), Euclid algor. biject. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
47 Let f F q [x 1,..., x k ] with k 2 Two notions: total deg. and vector deg. of f Problem Count # of irr. polys. of a given deg. in F q [x 1,..., x k ] How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
48 Let f F q [x 1,..., x k ] with k 2 Two notions: total deg. and vector deg. of f Problem Count # of irr. polys. of a given deg. in F q [x 1,..., x k ] Problem Count # of pairs of relatively prime polys. of a given deg. in F q [x 1,..., x k ] Each problem has a total deg. version and a vector deg. version How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
49 Hou/Mul. (FFA 09) Results for several variables How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
50 Hou/Mul. (FFA 09) Results for several variables Bodin (FFA 10) Generating series for # irr. of given deg. and for indecomp. polys. How Are Irreducible and Primitive Polynomials Distributed over July Finite 21, Fields? / 28
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