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A Factorization Theorem for Exponential Polynomials

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A Factorization Theorem for Exponential Polynomials A factorization theorem for exponential polynomials P. D'Aquino and G. Terzo Seconda Universit`adegli Studi di Napoli GOAL: We give a factorization theorem for the ring of exponential polynomials in many variables over an algebraically closed field of characteristic 0 with an expo- nentiation. 1 E-rings Definition 1. An exponential ring, or E-ring, is a pair (R; E) with R a ring (com- mutative with 1) and E :(R; +) ! (U(R); ·) a map of the additive group of R into the multiplicative group of units of R, satisfying 1. E(x + y) = E(x) · E(y) for all x; y 2 R 2. E(0) = 1: Example 1. (R; ax), with a > 0, and (C; ex). 2. (R; E) where R is any ring and E(x) = 1 for all x 2 R: 3. (S[t];E); where S is an E-field of characteristic 0 and S[t] is the ring of formal power series in t over S. Let f 2 S[t], where f = r + f1; and r 2 S; 1 X n E(f) = E(r) · (f1) =n! n=0 4. K[X]E ring of exponential polynomials over (K; E), an E-ring. (K; E) is an E-field if K is a field. 1 2 E-polynomial ring Let (K; E) be an E-field, the ring of E-polynomials in the indeterminates x = x1; : : : ; xn is an E-ring constructed as follows by recursion. • (Rk; +; ·)k≥−1 are rings; • (Bk; +)k≥0 are torsion free abelian groups, and for the algebraically closed fields, are also divisible groups; • (Ek)k≥−1 are partial E-morphisms. Step 0: Let R−1 = K; R0 = (K[x]; +; ·); B0 =< x >, the ideal generated by x. So R0 = R−1 ⊕ B0; E−1 : R−1 −! R0; is the composition of the initial E-morphism over K with the immersion of K into K[x]: Inductive step: Suppose k ≥ 0 and Rk−1, Rk, Bk and Ek−1 have been defined in such a way that: Rk = Rk−1 ⊕ Bk, Ek−1 :(Rk−1; +) ! (U(Rk); ·) U(Rk) = invertible elements of Rk. Let Bk t :(Bk; +) ! (t ; ·) be an isomorphism. Define Bk Rk+1 = Rk[t ] (group ring construction). • Rk is a subring of Rk+1 b • 1. Bk+1 is the Rk-submodule of Rk+1 freely generated by t where b 2 Bk, 2. Rk+1 = Rk ⊕ Bk+1 as additive groups Bk⊕Bk−1 Bk⊕Bk−1⊕Bk−2 Bk⊕:::⊕B0 • Rk+1 = Rk−1[t ] = Rk−2[t ] = :::::: = K[x][t ] b • Ek :(Rk; +) ! (U(Rk+1); ·) defined as follows Ek(x) = Ek−1(r) · t 2 Get a chain of partial E-rings (domain of Ek+1 = Rk) R0 ⊂ R1 ⊂ R2 · · · ⊂ Rk ⊂ · · · The ring of exponential polynomials is 1 E [ B0⊕:::⊕Bk⊕::: K[x] = lim Rk = Rk = K[x][t ] = U[G] k k=0 group ring where • U = K[x] a UFD • G = tB0⊕:::⊕Bn⊕::: a divisible torsion free abelian group (G is orderable and a Q-vector space) Exponentiation on K[x]E is defined as follows E(f) = Ek(f); if f 2 Rk and k 2 N: Proposition 1. If K is an exponential field of characteristic 0 then the E-polynomial ring K[x]E is an integral domain whose units are of the form E(α); where α 2 K[x]E. Definition 2. An exponential polynomial f 2 K[x]E is irreducible if there are no non-units g and h with f = gh: 3 Associate polynomial To any exponential polynomial we will associate a classical polynomial. E Let f 2 K[x] ; f 2 Rk for some k 2 N; then N X αh f = aht ; h=1 where ah 2 K[x] and αh 2 B0 ⊕ ::: ⊕ Bk−1: Definition 3. The support of f is the Q-vector space generated by α1; : : : ; αN ; and is denoted by supp(f). 3 • Let fν1; : : : ; νpg be a Q-base for supp(f). Then p X αi = rijνj j=1 for all i = 1;:::;N, and ri;j 2 Q: • Wlog we can assume rij 2 Z: Moreover, we can assume all rij's positive integers since we can multiply f by an invertible element, i.e. a purely exponential term. Then f is a polynomial in tν1 ; : : : ; tνp ; with coefficients in a UFD, U = K[x]: Let ν1 νp y1 = t ; : : : ; yp = t : If we substitute each αi by its expression in terms of the bases ν1; : : : ; νp we transform f into a classical polynomial in the variables y1; : : : ; yp E f 2 K[x] Q(y1; : : : ; yp) 2 U[y1; : : : ; yp] where U = K[x]. We will refer to Q(y1; : : : ; yp) as the associate polynomial of f. Definition 4. An exponential polynomial f is simple if dim(supp(f)) = 1. m1 mp In what follows a monomial in y1; : : : ; yp is a term y1 · ::: · yp with mi 2 Z: Definition 5. A classical polynomial Q(y1; : : : ; yp) is essentially 1-variable if there are monomials τ1; τ2 in y1; : : : ; yp such that Q(y1; : : : ; yp) = τ1P (τ2), where P is a polynomial in just one variable. Example: The polynomial 2 3 9 2 6 Q(x; y) = x y(3x y − 2x y + 1) = τ1P (τ2); 2 3 3 2 where τ1 = x y, τ2 = xy and P (z) = 3z − 2z + 1; is an essentially 1-variable. 4 Remarks: 1. The correspondence between f and Q holds modulo a monomial in y1; : : : ; yp which corresponds to an invertible element of K[x]E; 2. f is a simple polynomial iff Q is essentially 1-variable polynomial. Let Q(y1; : : : ; yp) 2 K[y1; : : : ; yp] be an irreducible polynomial over a UFD U: It µ1 µp can happen that for some µ1; : : : ; µp 2 N+;Q(y1 ; : : : ; yp ) becomes reducible. For example, Q(x; y) = x − y is irreducible, but Q(x3; y6) = (x − y2)(x2 + xy2 + y4). Definition 6. A polynomial Q(y1; : : : ; yp) is power irreducible over U if for each µ1 µp µ1; : : : ; µp 2 N+;Q(y1 ; : : : ; yp ) is irreducible. Definition 7. A polynomial Q(y1; : : : ; yp) is primary in yi if the g.c.d. of the expo- nents of yi in all terms of Q is 1. Definition 8. A polynomial Q(y1; : : : ; yp) is primary if it is primary in each vari- able. Example: • Q(x; y) = 3x2y − 5y3 + x3 is primary in both x and y: • R(x; y) = 3x2y − 5y3 + x4 is not primary in x since R(x; y) = P (x2; y), where P (x; y) = 3xy − 5y3 + x2 Remark: If Q(y1; : : : ; yp) is a non primary polynomial then there exists a unique p-upla t1; : : : ; tp of positive integers such that t1 tp Q(y1; : : : ; yp) = P (y1 ; : : : ; yp ) where P (y1; : : : ; yp) is primary. 5 4 Factorization Theorem Known results: • Ritt in [8] was the first to consider a factorization theory for exponential polynomials of the following form α1z αnz f(z) = a1e + ::: + ane ; where ai; αi 2 C: • Gourin and Macoll in [6], [7] gave refinements of Ritt's factorization theorem for exponential polynomials of the form α1z αnz f(z) = p1(z)e + ::: + pn(z)e ; where αi are complex numbers and pi(z) 2 C[z]. • Only in the mid '90s van der Poorten and Everest obtained a factorization theorem in a more general context for exponential polynomials of the second form over an algebraically closed field of characteristic 0 with exponentiation. We generalize the results obtained in [8], [3], [4] to any exponential polynomial with coefficients in an algebraically closed field K of characteristic 0 with exponentiation. Major issues for exponential polynomial: If fractional powers are permitted: 1. A binomial as y − 1 defined over an algebraically closed field K may have 1 infinitely many factors. Indeed, y k − , where is a kth root of unity, is a factor for any positive integer k: 2. The polynomial (x − y) becomes reducible 1 1 2 1 1 2 (x − y) = (x 3 − y 3 )(x 3 + x 3 y 3 + y 3 ): 6 There is a correspondence between a factorization of f and a factorization of Q in fractional powers of the variables. ()) It follows from: Lemma 1. Let f(x) 2 K[x]E and suppose that f(x) = g(x)·h(x); where g(x); h(x) 2 K[x]E: Then supp(g); supp(h) are contained in supp(f): For the proof we use the construction of the E-polynomial ring and the fact that G is an ordered group. Corollary 1. Suppose f factorizes as f = gh. Let Q, P and R be the associate polynomials of f, g and h, respectively. There is a monomial τ such that Q = P 0R0 where P 0 = P τ, R0 = τ −1R. (() It follows from: Theorem 1. Let Q(y1; : : : ; yp) be a (primary) irreducible polynomial over U, a unique factorization domain containing all roots of unity. Assume also that Q is not essentially a 1-variable polynomial. Then Q(y1; : : : ; yp) has a factorization into primary irreducible polynomials in the ring generated over U by all fractional powers of y1; : : : ; yp, and the number of these irreducible factors is finite. (t) t1 tp Proof: First of all notice that any factorization of Q = Q(y1 ; : : : ; yp ) for t1; : : : ; tp 2 N gives a factorization in fractional powers of the variables of Q(y1; : : : ; yp). t1 tp Hence we will study the factorizations of Q(y1 ; : : : ; yp ). Claim. There are only finitely many sets of positive integers t11; : : : ; t1p; t21; : : : ; t2p; :::::: ; tn1; : : : ; tnp ti1 tip such that Q(y1 ; : : : ; yp ), for i = 1; : : : ; n are reducible.
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