Towards a model theory for transseries by Matthias Aschenbrenner (UCLA) Lou van den Dries (Univ. of Illinois @ Urbana-Champaign) Joris van der Hoeven (CNRS, École polytechnique) Luminy, 08/06/2015 http://www.TeXmacs.org What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ ex + e ··· + ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ x ex + e + e x ··· + ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ x x ex + e + e + e x x 2 ··· + ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ x x x e e x e + x + 2 + 2 e + e x ··· + e ··· + log x ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ ex ex x ex + + + 2 ex + e + e x x 2 ··· + e x ··· + log x ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ x x x x ex + e + e + 2 ex + e + e + e x x 2 ··· + e x x 2 ··· + log x ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ x x x x x e e x e e e + x + 2 + 2 e + x + 2 + √x + e x ··· + e x ··· +e ··· + log x ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ ex ex ex ex ex + + + 2 ex + + + plog x +··· e x x 2 ··· + e x x 2 ··· +e√x +e + log x ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ x x x x x e e x e e plog log x +··· e + + + 2 e + + + x plog x +e e x x 2 ··· + e x x 2 ··· +e√ +e + log x ··· What is a transseries? 2/22 1 2 3456 78910111213141516171819 20 21 22 (x 1) ≻ x x x x x e e x e e plog log x +··· e + + + 2 e + + + x plog x +e e x x 2 ··· + e x x 2 ··· +e√ +e + log x ··· Dahn & Göring • Écalle • Detailed treatment in LNM 1888: “Transseries and Real Differential Algebra” • More examples 3/22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 1 2 e 3 e 1 − − = 1+ x − + x − + x − + x − + x − − + 1 x 1 x e − − ··· −x 1 1 1 x e 2x − − = 1+ + + +e− +2 + +e− + 1 x 1 +e x x x 2 x − ··· ··· ··· −x ex e = 1 1 + 2 6 + 24 120 + − x x − x 2 x 3 − x 4 x 5 − x 6 ··· R √ x (log x −1) √ x (log x −1) √ x (log x −1) Γ(x) = 2 p e + 2 p e + 2 p e + x 1/2 12 x 3/2 288 x 5/2 ··· x x x ζ(x) = 1+2− +3− +4− + ··· ϕ(x) = 1 + ϕ(x π)= 1 + 1 + 1 + 1 + x x x π x π2 x π3 ··· 1 log2 x 1 1 1 1 ψ(x) = + ψ(e )= + 2 + 4 + 8 + x x elog x elog x elog x ··· The exponential field T of transseries 4/22 123 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Transseries are transfinite series f = fm m (T: set of transmonomials, coefficients fm R) ∈ mXT ∈ ex ex ex x + + + π 1 1 = 3ee +e log x log2 x log3 x ··· x 11 +7 + + + + − − x x log x x(log x)2 ··· = f + f + f ≻ ≍ ≺ex ex ex x + + + f = 3ee +e log x log2 x log3 x ··· x 11 ≻ − − f = 7 ≍ π 1 1 f = + + + ≺ x x log x x(log x)2 ··· The exponential field T of transseries 4/22 123 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Transseries are transfinite series f = fm m (T: set of transmonomials, coefficients fm R) ∈ mXT ∈ ex ex ex x + + + π 1 1 = 3ee +e log x log2 x log3 x ··· x 11 +7 + + + + − − x x log x x(log x)2 ··· = f + f + f ≻ ≍ ≺ex ex ex x + + + f = 3ee +e log x log2 x log3 x ··· x 11 ≻ − − f = 7 ≍ π 1 1 f = + + + ≺ x x log x x(log x)2 ··· Transmonomials and exponential structure T = exp T ≻ T = f T: f = f ≻ { ∈ ≻ } exp f = (exp f )(exp f )(exp f ) ≻ ≍ ≺ 1 2 1 3 exp f = 1+ f + /2 f + /6 f + . ≺ ≺ ≺ ≺ ··· T as an ordered differential field 5/22 1234 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Differentiation x ′ = 1 f f (e )′ = f ′ e fm m ′ = fm m′ mXT mXT ∈ ∈ T as an ordered differential field 5/22 1234 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Differentiation x ′ = 1 f f (e )′ = f ′ e fm m ′ = fm m′ mXT mXT ∈ ∈ Ordering f > 0 ( g T, g 2 = f ) c(f ) > 0 ⇐⇒ ∃ ∈ ⇐⇒ ex ex ex x + + + π 1 c 3 ee +e log x log2 x log3 x ··· x 11 +7+ + + = 3 − − x x log x ··· − T as an ordered differential field 5/22 1234 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Differentiation x ′ = 1 f f (e )′ = f ′ e fm m ′ = fm m′ mXT mXT ∈ ∈ Ordering f > 0 ( g T, g 2 = f ) c(f ) > 0 ⇐⇒ ∃ ∈ ⇐⇒ ex ex ex x + + + π 1 c 3 ee +e log x log2 x log3 x ··· x 11 +7+ + + = 3 − − x x log x ··· − Valuation = f T: c R>, f 6 c O { ∈ ∃ ∈ | | } = f T: c R>, f < c O { ∈ ∀ ∈ | | } / = R O O ∼ T as an ordered differential field 5/22 1234 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Differentiation x ′ = 1 f f (e )′ = f ′ e fm m ′ = fm m′ mXT mXT ∈ ∈ Ordering f > 0 ( g T, g 2 = f ) c(f ) > 0 ⇐⇒ ∃ ∈ ⇐⇒ ex ex ex x + + + π 1 c 3 ee +e log x log2 x log3 x ··· x 11 +7+ + + = 3 − − x x log x ··· − Valuation = f T: c R>, f 6 c O { ∈ ∃ ∈ | | } = f T: c R>, f < c O { ∈ ∀ ∈ | | } / = R O O ∼ v(f ) > v(g) f 4 g f g v(f ) > v(g) ⇐⇒ f g ⇐⇒ f ∈Og ⇐⇒ ≺ ⇐⇒ ∈ O Remarkable closure properties 6/22 12345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Basic closure properties T is Liouville closed (i.e. real closed, stable under integration and exponential integration). Remarkable closure properties 6/22 12345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Basic closure properties T is Liouville closed (i.e. real closed, stable under integration and exponential integration). Factorization of linear differential operators [vdH] Any linear differential operator L T[∂] can be factored into operators of order at most two. ∈ Remarkable closure properties 6/22 12345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Basic closure properties T is Liouville closed (i.e. real closed, stable under integration and exponential integration). Factorization of linear differential operators [vdH] Any linear differential operator L T[∂] can be factored into operators of order at most two. ∈ Differential intermediate value theorem [vdH] (r ) Given a differential polyonomial P T Y (i.e. P = P(Y , Y ′, ..., Y )) and f < g in T such that P(f ) P(g) < 0, there exists an∈y { T}with f < y < g and P(y)=0. ∈ Remarkable closure properties 6/22 12345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Basic closure properties T is Liouville closed (i.e. real closed, stable under integration and exponential integration). Factorization of linear differential operators [vdH] Any linear differential operator L T[∂] can be factored into operators of order at most two. ∈ Differential intermediate value theorem [vdH] (r ) Given a differential polyonomial P T Y (i.e. P = P(Y , Y ′, ..., Y )) and f < g in T such that P(f ) P(g) < 0, there exists an∈y { T}with f < y < g and P(y)=0. ∈ Newtonianity [vdH], differential analogue of henselian fields Any quasi-linear differential equation with asymptotic side-condition P(y)=0, y v ≺ admits a solution (P T Y , v T ). ∈ { } ∈ ∪{∞} Remarkable closure properties 6/22 12345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Basic closure properties T is Liouville closed (i.e. real closed, stable under integration and exponential integration). Factorization of linear differential operators [vdH] Any linear differential operator L T[∂] can be factored into operators of order at most two. ∈ Differential intermediate value theorem [vdH] (r ) Given a differential polyonomial P T Y (i.e. P = P(Y , Y ′, ..., Y )) and f < g in T such that P(f ) P(g) < 0, there exists an∈y { T}with f < y < g and P(y)=0. ∈ Newtonianity [vdH], differential analogue of henselian fields Any quasi-linear differential equation with asymptotic side-condition P(y)=0, y v ≺ admits a solution (P T Y , v T ). ∈ { } ∈ ∪{∞} H-fields 7/22 123456 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Definition.