GENERATING SERIES FOR IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS

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1 GENERATING SERIES FOR IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS ARNAUD BODIN Abstract. We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multi-degree and the case of indecomposable polynomials.. Introduction A well-nown formula that goes bac to Gauss estimate the number I n of monic irreducible polynomials among all the N n monic polynomials of degree n in F q [x]: I n N n n. A proof of this fact from the combinatorial point of view has been formalized by Ph. Flajolet, X. Gourdon and D. Panario in [6]. They apply this formalism to count irreducible polynomials in one variable over F q with the help of generating series. These generating series are convergent. The goal of this paper is to count irreducible polynomials in several variables. While the formalism is the same, the series are now formal power series. At a first glance it seems a major obstruction, but despite that the series are non-convergent lots of manipulations are possible: computation of the terms of a generating series, approximation of this term, Möbius inversion formula,... For comments and examples in this vein we refer to the boo of Ph. Flajolet and R. Sedgewic [7] (and especially Examples I.9, I.20, II.5, Note II.28 and Appendix A). We start by extending the introduction and recall in Section 2 the formalism introduced in [6], we consider in Section 3 the case of irreducible polynomials in several variables of a given degree. Then in Section 4 the irreducible polynomials are counted with respect to the multi-degree (or vector-degree). Finally in Section 5 we deal with indecomposable polynomials. In each case we give an exact formula, a generating series, an asymptotic development at first order and end by Date: October 6, Mathematics Subject Classification. 2E05, T06. Key words and phrases. Irreducible and indecomposable polynomials, Inversion formula.

2 2 ARNAUD BODIN an example. References to earlier wors are given for each cases after the main statements. Acnowledgments. I am pleased to than Philippe Flajolet: this article starts after one of his question during a visit at inria Rocquencourt, he also explained me how non-convergent series can be nicely handled. 2. Formalism Let us recall the formalism of [6]. Let I be a family of primitive combinatorial objects and ω I, then = + ω + ω ω2 + formally represents arbitrary sequences composed of ω. The product N = ω I formally represents all multisets (sets with possible ω repetitions) composed of elements of I. In our situation I will be the set of all monic irreducible polynomials and N the set of all monic polynomials. The relation N = ω I ω expresses that each monic polynomial admits a unique decomposition into monic irreducible polynomials. In order to count polynomials over F q we mae the following substitution ω z deg ω, where z is a new formal variable and deg ω is the degree of the polynomial ω. We get the generating series: N(z) = ω N z deg ω = n N n z n, I(z) = ω I z deg ω = n I n z n, that count the total number of monic polynomials and monic irreducible polynomials. The decomposition of polynomials into irreducible factors gives the relation: + N(z) = z = deg ω ( z n ). In ω I n By taing the logarithm we get: log( + N(z)) = n I n log( z n ) = n I(z n ) n. For one variable polynomials we now that the number of monic polynomials is N n = q n so that + N(z) = + n qn z n =. qz Hence () log qz = n I(z n ) n, by applying Möbius inversion formula to the (convergent) series we get I(z) = n µ(n) n log qz.

3 IRREDUCIBLE POLYNOMIALS 3 If we prefer, we can only consider the coefficients of the terms z n in Equation () to find q n n = I n/ and I n = µ()q n/ n n n by applying the classical Möbius inversion formula (µ is Möbius ( function). From this last expression, we deduce that I n = qn + O q n/2, ) n n that implies the first statement of the introduction. 3. Irreducible polynomials: degree Let ( ) ν + n b n = and N n = qbn q b n. ν q Then N n is the number of irreducible, monic, polynomials of degree (exactly) n in F q [x,..., x ν ]. The corresponding generating series is N(z) = n N n z n. Let I n be the number irreducible monic polynomials of degree n in F q [x,..., x ν ], we define the corresponding generating series: I(z) = n I n z n. Theorem. () I(z) = µ(n) n log( + n N(zn )), (2) I n = µ() n [z n ] log( + N(z)), (3) I n = N n N N n + O(q b n 2+b 2 2 ). In item (3), ν and q are fixed while the error term O is for values of n that tend to infinity. The history for item (3) starts with L. Carlitz [3] who proved that almost all polynomials in several variables are irreducible. His wor has been extended to the case of the bi-degree in [4] (as considered in Section 4) and by S. Cohen [5] for more variables. Here we get that most of polynomials are irreducible and an estimation of the error term. A similar formula has been obtained in []: In N n N N n N n as n grows to infinity. A formula with explicit constants is given in [8] for polynomials in two variables In N n q n+ 2q n, with n 6. An asymptotic formula of higher order is proved in []: I n = N n + α N n + α 2 N n α s N n s + O(N n s ), where α i are constants and the order s is arbitrary large. Our proof of (3) appears to be simple compare with these references. Finally, in the forthcoming [0], the formula I(z) is also obtained and applied to get an approximation: I n = N n q bn+ν ( + 2/q + O(/q 2 )), where ν and n 5 are fixed, while q grows to infinity.

4 4 ARNAUD BODIN The end of this section is devoted to the proof of Theorem. Proof. Generating series. We follow the formalism of the introduction: Hence: + N(z) = n (2) L(z) := log( + N(z)) = n ( z n ) In. I n log( z n ) = n I(z n ) n. By Möbius inversion formula applied to the series (see comments after the proof) we get: I(z) = µ(n) n L(zn ). n Another point of view is to mae computations term by term: we tae the coefficients of z n in Equation (2) and get [z n ]L(z) = n I n/. Then we apply the usual Möbius inversion formula to n[z n ]L(z) and find (3) I n = µ() [z n ]L(z). n Asymptotic behaviour. We start with the computation of an approximation of N(z) 2. [z n ]N(z) 2 = [z n ] ( N z + N 2 z 2 + )2 = 2N n N + O(q b n 2+b 2 2 ). And for powers 3 we have: [z n ]N(z) = O(q b n 2+b 2 2 ). Then we compute the asymptotic behaviour of [z n ]L(z). [z n ]L(z) = [z n ] log( + N(z)) = [z n ]N(z) 2 [zn ]N(z) 2 + = N n ( 2 N N n + O(q b n 2+b 2 2 ) ) + O(q b n 2+b 2 2 ) 2 = N n N N n + O(q b n 2+b 2 2 ).

5 Now we get: I n = n IRREDUCIBLE POLYNOMIALS 5 µ() [z n ]L(z) = µ() [zn ]L(z) + n,> µ() [z n ]L(z) = N n N N n + O(q b n 2+b 2 2 ) + O(q b n 2 ) = N n N N n + O(q b n 2+b 2 2 ). Example. For example, in two variables (ν = 2), we can compute the exact expression of I 00. We compute the first 00 coefficients of the expansion of L(z) = log( + N(z)) into powers of z (by derivation an expansion of L(z) can be obtained from the inverse of + N(z)). The exact value of I 00 is then computed from formula (2) of Theorem : I 00 = q (q55 q 5052 q 505 q 5050 q q q ) which has a total of 4385 monomials! When we specialize the approximation of Theorem to n = 00 we find that I 00 is about q (q55 q 5052 q 505 q 5050 ) with an error term of magnitude q Finally if we set q = 2 in the exact expression, we get a number with 55 digits. I 00 = , Inversion formula. We shall now setch the proof of inversion formula for formal power series. Suppose that f, g K[[z]] with f(0) = 0 and g(z) = f(z ) = f(z) + f(z 2 ) + f(z 3 ) + then f(z) = n µ(n)g(z n ) = µ()g(z) + µ(2)g(z 2 ) + µ(3)g(z 3 ) + where µ(n) is the ordinary Möbius function. The proof is in fact the same as the classic one: µ(n)f(z n ). n µ(n)g(z n ) = n In this expression the coefficient of f(z i ) involves only a finite number of terms and is j i µ(j) = so that the former sum becomes: f(z i ) µ(j) = f(z i ). i j i i

6 6 ARNAUD BODIN Further readings for comments on computations with formal power series can be found in [7, Appendix A]. For convergent series, see [3, p. 98] and for a general version of inversion theorem, see [4]. 4. Irreducible polynomials: multi-degree Let N n be the number of irreducible, monic, polynomials of multidegree n = (n,..., n ν ) in F q [x,..., x ν ]. By [5] or [] we have N n = ( ) ν+δ+ +δν q (n+δ) (nν+δν). q (δ,...,δ ν) {0,} ν The corresponding generating series is N(z,..., z ν ) = N n,...,n ν z n zν nν, or for short (n,...,n ν) (0,...,0) N(z) = n 0 N n z n. Let I n be the number of irreducible monic polynomials of degree n in F q [x,..., x ν ], we define the corresponding generating series: I(z) = n 0 I n z n. Theorem 2. µ() () I(z) = log( + N(z )), (2) I n = µ() gcd(n) [z n ] log( + N(z)), (3) I n = N n N,0,...,0 N n,n 2,...,n ν +O(N 0,,0,...,0 N n,n 2,n 3,...,n ν )+ O(N 2,0,...,0 N n 2,n 2,...,n ν ) if n n 2 n ν. For (3) X.-D. Hou and G. Mullen in [] have already obtained a similar formula. The scheme of the proof is the same as in the previous section. Proof. Generating series. The proof is an application of the formalism of the introduction, (used in a similar way as in Section 3, see also [4] heuristic proof of (2.)). + N(z) = n 0 ( z n ) In. Hence: (4) L(z) := log( + N(z)) = n 0 I n log( z n ) = I(z ),

7 IRREDUCIBLE POLYNOMIALS 7 where z = (z,..., zν). Now tae the coefficient of z n in Equation (4), we get [z n ]L(z) = I n/. gcd(n) By Möbius formula applied to gcd(n)[z n ]L(z) we get I n = µ() [z n ]L(z). gcd(n) We can also directly apply Möbius inversion formula to L(z): I(z) = µ() L(z ). Asymptotic behaviour. We suppose n n 2 n ν. We have: I n = µ() [z n ]L(z) gcd(n) = µ() [zn ]L(z) + gcd(n),> µ() [z n ]L(z) =N n 2( 2N,0,...,0 N n,n 2,...,n ν + 2N 0,,0,...,0 N n,n 2,n 3,...,n ν + + 2N 0,...,0, N n,n 2,...,n ν,n ν + O(N,,,0,...,0 N n,n 2,n 3,...,n ν ) ) + O(N 2,0,...,0 N n 2,n 2,...,n ν ) =N n N,0,...,0 N n,n 2,...,n ν + O(N 0,,0,...,0 N n,n 2,n 3,...,n ν )+ + O(N 2,0,...,0 N n 2,n 2,...,n ν ). Intuitively these formulas express that most of reducible polynomials are of type P (x,..., x ν ) = (x + α) Q(x,..., x ν ), where deg x Q = (deg x P ), α being a constant. Example. For example, we get I,5 = q (q72 q 67 q 66 q 60 + ), while N,5 = q (q72 q 66 q 60 + q 55 ). The approximation of Theorem 2 specialized to (, 5) gives that I,5 is about q (q72 q 67 q 66 ) with an error term of magnitude q 6. Finally if we fix q = 2, we get I,5 = and N,5 = Indecomposable polynomials Definition and nown facts. A polynomial P in F q [x,..., x ν ] is decomposable if there exist h F q [t] with deg h 2 and Q F q [x,..., x ν ] such that P (x,..., x ν ) = h Q(x,..., x ν ).

8 8 ARNAUD BODIN Otherwise P is said to be indecomposable. For example P (x, y) = x 3 y 3 +2xy+ is decomposable (with h(t) = t 3 +2t+ and Q(x, y) = xy). Notice that if P c is irreducible in F q [x,..., x ν ] for some value c F q then P is indecomposable. Indecomposable polynomials have a special interest in regards of Stein s theorem and provide families of irreducible polynomials (see [5], [2], [2]): Theorem (Stein s theorem). If P F q [x,..., x ν ] is indecomposable then for all but finitely many values c F q (the algebraic closure of F q ), P c is irreducible in F q [x,..., x ν ]. Moreover the number of values c such that P c is reducible is strictly less than deg P. Let J n be the number of indecomposable polynomials of degree (exactly) n in F q [x,..., x ν ], ν 2. Let N n be the number of polynomials of degree n (not necessarily monic) in F q [x,..., x ν ]: N n = q bn q b n. We define the following Dirichlet series: N(s) = n N n n s, J(s) = n J n n s, F (s) = n q n n s. We also define a ind of Möbius function µ(d, n): µ(d, n) = ( ) q d d + d d, d=d d 2 d 3 d d + =n, d <d 2 < <d <n where the sum is over all the iterated divisors of n ( not being fixed). We define µ(n, n) =. For a fixed n, we define < l < l to be the first two divisors of n. Theorem 3. () J(s) = N(s), F (s) (2) J n = d n, d n µ(d, n) N d, (3) If n is the product of at least three prime numbers: J n = N ( ) n q l N n + O q l+l 2 N n. l l The item (3) has already been obtained in [2] and [9], where some explicit constants are computed. For example in two variables, and when n is the product of at least three prime numbers, Theorem 5. of [2] yields the same first order estimate of J n : N n J n q l q N n n l q n/l n ql q n/l N n l q n q n/l.

9 IRREDUCIBLE POLYNOMIALS 9 Proof. Generating series and inversion formula. The decomposition of a decomposable polynomial is unique once it has been normalized, see [2, Section 5]: Lemma 4. Let P = h Q be a decomposition with deg h 2, Q indecomposable, monic and its constant term equals zero. Then such a decomposition is unique. By induction it implies the following lemma (see [2]): Lemma 5. J n = N n From Lemma 5 we get: N n = d n, d<n d n, d n q n d J d. q n d J d. From the point of view of Dirichlet series it yields: N(s) = J(s) F (s). Then by induction we also deduce from Lemma 5 that J n = µ(d, n) N d. d n, d n Asymptotic behaviour. From this formula we deduce: J n = µ(n, n) N n + µ (n/l, n) N ( n + O µ (n/l, n) N ) n l l = N ( n q l N n + O (q l+l 2 q l ) N ) n l l ( ) = N n q l N n + O q l+l 2 N n. l l Example. For example, in two variables, we have N 00 = q 55 q 5050 and we compute J 00 by formula (2) of Theorem 3. We get J 00 = q 55 q 5050 q q 276 q While the approximation given in Theorem 3 is that J 00 is about q 55 q 5050 q q 276 with an error term of magnitude q 355. References [] A. Bodin, Number of irreducible polynomials in several variables over finite fields, American Math. Monthly 5 (2008), [2] A. Bodin, P. Dèbes, S. Najib, Indecomposable polynomials and their spectrum, Acta Arith. 39 (2009), [3] L. Carlitz, The distribution of irreducible polynomials in several indeterminates, Illinois J. Math. 7 (963),

10 0 ARNAUD BODIN [4] L. Carlitz, The distribution of irreducible polynomials in several indeterminates. II, Canad. J. Math. 7 (965), [5] S. Cohen, The distribution of irreducible polynomials in several indeterminates over a finite field, Proc. Edinburgh Math. Soc. 6 (968/969) 7. [6] Ph. Flajolet, X. Gourdon, D. Panario, The complete analysis of a polynomial factorization algorithm over finite fields, J. Algorithms 40 (200), [7] Ph. Flajolet, R. Sedgewic, Analytic Combinatorics, Cambridge University Press, [8] J. von zur Gathen, Counting reducible and singular bivariate polynomials, Finite Fields Appl. 4 (2008), [9] J. von zur Gathen, Counting decomposable multivariate polynomials, preprint. [0] J. von zur Gathen, A. Viola, K. Ziegler, Counting reducible, squareful, and relatively irreducible multivariate polynomials over finite fields, preprint. [] X.-D. Hou, G. Mullen, Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields, Finite Fields Appl. 5 (2009), [2] S. Najib, Une généralisation de l inégalité de Stein-Lorenzini, J. Algebra 292 (2005), [3] N. Negoescu, Generalizations of the Mobius theorem of inversion, Timosoara Editura Waldpress, 995 [4] G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlicheitstheorie und Verw. Gebiete 2 (964), [5] Y. Stein, The total reducibility order of a polynomial in two variables, Israel J. Math. 68 (989), address: Arnaud.Bodin@math.univ-lille.fr Laboratoire Paul Painlevé, Mathématiques, Université Lille, Villeneuve d Ascq Cedex, France