The construction of the field of Transseries − a logarithmic-exponential field extension of R((x 1)) Santiago Camacho on work by van den Dries, Macintyre, and Marker Santiago Camacho 1/ 29 Logarithmic Exponential Field Definition A exponential field is a tuple (K ; E), where K is an ordered field and E is a strictly increasing group homomorphism from E :(K ; +) ! (K >0; ×) In principle E need not be surjective. but if E(K ) = K >0 we call (K ; E) a logarithmic-exponential field. A familiar example is (R; exp) Santiago Camacho 2/ 29 Hahn Field Let M be an ordered monomial multiplicative group and K an ordered field. Consider the set ff 2 KM : supp(f ) is noetherian g. We will write its elements as X X f (m)m = fmm; m2M m2M We will denote by Lm(f ) = maxfsupp(f )g the leading monomial of f and by Lc(f ) = fLm(f ) its leading coefficient. Santiago Camacho 3/ 29 and product given by ! X X X X fmm · gmm = (fm1 gm2 ) m m2M m2M m2M m1m2=m With this operations we have constructed a ring, K[[M]]. One checks that in general it is a field. Together with the ordering f > 0 , Lc(f ) >K 0 we have constructed an ordered field. We endow the above mentioned set with pointwise addition X X X fmm + gmm = (fm + gm)m; m2M m2M m2M Santiago Camacho 4/ 29 With this operations we have constructed a ring, K[[M]]. One checks that in general it is a field. Together with the ordering f > 0 , Lc(f ) >K 0 we have constructed an ordered field. We endow the above mentioned set with pointwise addition X X X fmm + gmm = (fm + gm)m; m2M m2M m2M and product given by ! X X X X fmm · gmm = (fm1 gm2 ) m m2M m2M m2M m1m2=m Santiago Camacho 4/ 29 We endow the above mentioned set with pointwise addition X X X fmm + gmm = (fm + gm)m; m2M m2M m2M and product given by ! X X X X fmm · gmm = (fm1 gm2 ) m m2M m2M m2M m1m2=m With this operations we have constructed a ring, K[[M]]. One checks that in general it is a field. Together with the ordering f > 0 , Lc(f ) >K 0 we have constructed an ordered field. Santiago Camacho 4/ 29 Given a family (fi )i2I with fi 2 K[[M]], it makes sense to talk about X fi i2I as long as for each m 2 M the set fi 2 I : m 2 supp(fi )g is finite, S and supp(fi ) is noetherian. Santiago Camacho 5/ 29 Example Z k k1 k2 Let M = x = fx : k 2 Zg with the order x > x , k1 > k2 K = R. x−n Then [[M]] = ((x−1)), and for the family we have R R n! n2N X x−n 2 [[xZ]]. n! R n2N − If we are to get a logarithmic exponential extention of R((x 1)) we would have for r 2 R E(log(rx)) = xr : Hence it is natural to start with M = xR ordered in the natural way. Santiago Camacho 6/ 29 Pre-exponential field X f n In [[xR]] it is natural to define E(f ) = but only when R n! n2N Lm(f ) < 1 = x0. The same construction does not work for f = x. Definition (K ; A; B; E) is called a pre-exponential field if 1 K is an ordered field, 2 A is an additive subgroup, 3 B is a convex subgroup, 4 K = A ⊕ B, >0 5 E : B ! K is an order-preserving group homomorphism. Santiago Camacho 7/ 29 e.g. x2 + x3=2 + x4=3 + ··· + x + 3 + 2x−π + x−4 + ::: B = ff 2 K : Lm(f ) ≤ 1g [ f0g e.g. 3 + 2x−π + x−4 + ::: A = ff 2 K : supp(f ) > 1g [ f0g e.g. x2 + x3=2 + x4=3 + ··· + x >0 P n E : B ! K with E(r + ) = exp(r) n! . Note that B in this case is a local ring. Example K = R[[xR]] Santiago Camacho 8/ 29 B = ff 2 K : Lm(f ) ≤ 1g [ f0g e.g. 3 + 2x−π + x−4 + ::: A = ff 2 K : supp(f ) > 1g [ f0g e.g. x2 + x3=2 + x4=3 + ··· + x >0 P n E : B ! K with E(r + ) = exp(r) n! . Note that B in this case is a local ring. Example K = R[[xR]] e.g. x2 + x3=2 + x4=3 + ··· + x + 3 + 2x−π + x−4 + ::: Santiago Camacho 8/ 29 e.g. 3 + 2x−π + x−4 + ::: A = ff 2 K : supp(f ) > 1g [ f0g e.g. x2 + x3=2 + x4=3 + ··· + x >0 P n E : B ! K with E(r + ) = exp(r) n! . Note that B in this case is a local ring. Example K = R[[xR]] e.g. x2 + x3=2 + x4=3 + ··· + x + 3 + 2x−π + x−4 + ::: B = ff 2 K : Lm(f ) ≤ 1g [ f0g Santiago Camacho 8/ 29 A = ff 2 K : supp(f ) > 1g [ f0g e.g. x2 + x3=2 + x4=3 + ··· + x >0 P n E : B ! K with E(r + ) = exp(r) n! . Note that B in this case is a local ring. Example K = R[[xR]] e.g. x2 + x3=2 + x4=3 + ··· + x + 3 + 2x−π + x−4 + ::: B = ff 2 K : Lm(f ) ≤ 1g [ f0g e.g. 3 + 2x−π + x−4 + ::: Santiago Camacho 8/ 29 e.g. x2 + x3=2 + x4=3 + ··· + x >0 P n E : B ! K with E(r + ) = exp(r) n! . Note that B in this case is a local ring. Example K = R[[xR]] e.g. x2 + x3=2 + x4=3 + ··· + x + 3 + 2x−π + x−4 + ::: B = ff 2 K : Lm(f ) ≤ 1g [ f0g e.g. 3 + 2x−π + x−4 + ::: A = ff 2 K : supp(f ) > 1g [ f0g Santiago Camacho 8/ 29 >0 P n E : B ! K with E(r + ) = exp(r) n! . Note that B in this case is a local ring. Example K = R[[xR]] e.g. x2 + x3=2 + x4=3 + ··· + x + 3 + 2x−π + x−4 + ::: B = ff 2 K : Lm(f ) ≤ 1g [ f0g e.g. 3 + 2x−π + x−4 + ::: A = ff 2 K : supp(f ) > 1g [ f0g e.g. x2 + x3=2 + x4=3 + ··· + x Santiago Camacho 8/ 29 Note that B in this case is a local ring. Example K = R[[xR]] e.g. x2 + x3=2 + x4=3 + ··· + x + 3 + 2x−π + x−4 + ::: B = ff 2 K : Lm(f ) ≤ 1g [ f0g e.g. 3 + 2x−π + x−4 + ::: A = ff 2 K : supp(f ) > 1g [ f0g e.g. x2 + x3=2 + x4=3 + ··· + x >0 P n E : B ! K with E(r + ) = exp(r) n! . Santiago Camacho 8/ 29 Example K = R[[xR]] e.g. x2 + x3=2 + x4=3 + ··· + x + 3 + 2x−π + x−4 + ::: B = ff 2 K : Lm(f ) ≤ 1g [ f0g e.g. 3 + 2x−π + x−4 + ::: A = ff 2 K : supp(f ) > 1g [ f0g e.g. x2 + x3=2 + x4=3 + ··· + x >0 P n E : B ! K with E(r + ) = exp(r) n! . Note that B in this case is a local ring. Santiago Camacho 8/ 29 We need to extend E to all of K = A ⊕ B, so we need to figure out what E is on A. Consider E(A) = fE(a): a 2 Ag where E(a)E(b) = E(a + b) and E(a) < E(b) if a < b. A multiplicative copy of A. in a way that EA : A ! E(A), where EA(a) = E(a), is an order preserving group homomorphism from A to E(A). Note that E(A) is a monomial multiplicative ordered group. Santiago Camacho 9/ 29 The first pre-exponential extension By setting K 0 = K [[E(A)]], B0 = ff 2 K 0 : Lm(f ) ≤ 1 = E(0)g [ f0g, A0 = ff 2 K 0 : supp(f ) > 1 = E(0)g [ f0g, and E0 : B0 −! K 0>0 defined by X n E0(a + b + ) = E (a)E(b) A n! n2N we get that (K 0; A0; B0; E0) is a pre-exponential field extension of (K ; A; B; E) where K ⊆ Dom(E0). Santiago Camacho 10/ 29 Santiago Camacho 11/ 29 We shall go to infinity! (a particular construction actually goes beyond but this lies out of the current scope). R With K0 = K = R[[x ]] we construct K1 = K0[[E(A)]] K2 = K1[[E(A1)]] . Note that Kn+1 = Kn[[E(An)]] = Kn−1[[E(An−1)]][[E(An)]] can be identified with Kn−1[[E(An−1)E(An)]] Unfortunately E0 is not defined everywhere. For example what would E0(E0(x)) be? Santiago Camacho 12/ 29 (a particular construction actually goes beyond but this lies out of the current scope). R With K0 = K = R[[x ]] we construct K1 = K0[[E(A)]] K2 = K1[[E(A1)]] . Note that Kn+1 = Kn[[E(An)]] = Kn−1[[E(An−1)]][[E(An)]] can be identified with Kn−1[[E(An−1)E(An)]] Unfortunately E0 is not defined everywhere. For example what would E0(E0(x)) be? We shall go to infinity! Santiago Camacho 12/ 29 Note that Kn+1 = Kn[[E(An)]] = Kn−1[[E(An−1)]][[E(An)]] can be identified with Kn−1[[E(An−1)E(An)]] Unfortunately E0 is not defined everywhere. For example what would E0(E0(x)) be? We shall go to infinity! (a particular construction actually goes beyond but this lies out of the current scope). R With K0 = K = R[[x ]] we construct K1 = K0[[E(A)]] K2 = K1[[E(A1)]] .