Hardy Fields, Transseries, And Surreal Numbers

Hardy Fields, Transseries, and Surreal NumbersLou van den DriesUniversity of Illinois at Urbana-ChampaignPAULO RIBENBOIM DAY at IHP, March 20, 2018Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20181/ 16

IntroductionThis concerns joint and ongoing work with Matthias Aschenbrenner and Joris van der Hoeven.The three topics in the title are intimately related. In all three contexts we deal with valueddifferential fields, and the value groups are typically very large. Thus valuation theory asrepresented in Paulo’s Théorie des valuations plays a key role.Our book Asymptotic Differential Algebra and Model Theory of Transseries appeared last yearin the Annals of Mathematics Studies (Princeton University Press). It is full of constructionsinvolving pseudocauchy sequences. In our ongoing work based on it we also need the notion ofstep-complete (“complet-par-étages”) and its properties, which we learned from Paulo.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20182/ 16

IntroductionWe first discuss (maximal) Hardy fields, where our results are partly still conjectural.We hope to prove our conjectures in about a year from now.Next I discuss some results on the valued differential field T of transseries from ourbook. We didn’t consider there the conjectured relation to maximal Hardy fields.Two years ago, Berarducci and Mantova were able to equip Conway’s field of surrealnumbers with a natural and in some sense simplest possible derivation. Using resultsfrom our book we established a strong connection of the resulting valued differentialfield to T and to Hardy fields. This will be discussed in the last part of my talk.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20183/ 16

Hardy FieldsExamples of Hardy fields: Q, R, R(x), R(x, ex ), R(x, ex , log x).The elements of a Hardy field are germs at of differentiable real valued functions. AHardy field is closed under taking derivatives.To be precise, let C 1 be the ring of germs at of continuously differentiable real valuedfunctions defined (at least) on an interval (a, ). Then a Hardy field is according toBourbaki a subring H of C 1 such that H is a field that contains with each germ of a function falso the germ of its derivative f 0 (where f 0 might be defined on a smaller interval than f ).We denote the germ at of a function f also by f , relying on context.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20184/ 16

Hardy fields, continuedLet H be a Hardy field.Hardy fields are ordered fields: for f H, either f (t) 0 eventually, or f (t) 0,eventually, or f (t) 0, eventually; this is because f 6 0 in H implies f has an inverse in H, sof cannot have arbitrarily large zeros.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20185/ 16

Hardy fields, continuedLet H be a Hardy field.Hardy fields are ordered fields: for f H, either f (t) 0 eventually, or f (t) 0,eventually, or f (t) 0, eventually; this is because f 6 0 in H implies f has an inverse in H, sof cannot have arbitrarily large zeros.Hardy fields are valued fields: for f , g H, f 4 g means that for some positive constant cwe have f (t) 6 c g (t) , eventually. This is equivalent to v (f ) v (g ) for the naturalvaluation v on H.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20185/ 16

Hardy fields, continuedLet H be a Hardy field.Hardy fields are ordered fields: for f H, either f (t) 0 eventually, or f (t) 0,eventually, or f (t) 0, eventually; this is because f 6 0 in H implies f has an inverse in H, sof cannot have arbitrarily large zeros.Hardy fields are valued fields: for f , g H, f 4 g means that for some positive constant cwe have f (t) 6 c g (t) , eventually. This is equivalent to v (f ) v (g ) for the naturalvaluation v on H.Hardy fields are differential fields: this speaks for itself. For f in H, there are three cases:f 0 0, so f is eventually strictly decreasing;f 0 0, so f is eventually constant;f 0 0, so f is eventually strictly increasing.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20185/ 16

Extending Hardy fieldsHere are some basic extension results on Hardy fields H:H has a unique algebraic Hardy field extension that is real closedif h H, then eh generates a Hardy field H(eh )Rany antiderivative g h of any h H generates a Hardy field H(g )Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20186/ 16

Extending Hardy fieldsHere are some basic extension results on Hardy fields H:H has a unique algebraic Hardy field extension that is real closedif h H, then eh generates a Hardy field H(eh )Rany antiderivative g h of any h H generates a Hardy field H(g )Special cases of the last item: H(R) and H(x) are Hardy fields, and if h H , then H(log h)is a Hardy field. Thus maximal Hardy fields contain R, are real closed, and closed underexponentiation and integration. (Zorn guarantees the existence of maximal Hardy fields; thereare at least continuum many different maximal Hardy fields.)Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20186/ 16

A conjecture about maximal Hardy fieldsOur work in progress (ADH) has as its main goal to prove the following intermediate valueproperty for differential polynomials P(Y ) H[Y , Y 0 , Y 00 , .] over Hardy fields H:Whenever f g in H and P(f ) 0 P(g ), then P(y ) 0 for some y in some Hardyfield extension of H with f y g .Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20187/ 16

A conjecture about maximal Hardy fieldsOur work in progress (ADH) has as its main goal to prove the following intermediate valueproperty for differential polynomials P(Y ) H[Y , Y 0 , Y 00 , .] over Hardy fields H:Whenever f g in H and P(f ) 0 P(g ), then P(y ) 0 for some y in some Hardyfield extension of H with f y g .Equivalently, maximal Hardy fields have the intermediate value property for differentialpolynomials. The conjecture implies that all maximal Hardy fields are elementarily equivalent.(This implication depends on deep results to be discussed later in connection with transseries.)We have a roadmap for establishing the conjecture and have gone maybe a third of the way,but it might easily take another year to arrive at the finish line.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20187/ 16

Another conjecture about Hardy fieldsA secondary goal is to show that maximal Hardy fields are η1 -sets, using Hausdorff’sterminology about totally ordered sets. Equivalently:For any Hardy field H and countable sets A B in H we have A y B for some y insome Hardy field extension of H.Assuming CH, the two conjectures together imply that all maximal Hardy fields are isomorphic.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20188/ 16

Another conjecture about Hardy fieldsA secondary goal is to show that maximal Hardy fields are η1 -sets, using Hausdorff’sterminology about totally ordered sets. Equivalently:For any Hardy field H and countable sets A B in H we have A y B for some y insome Hardy field extension of H.Assuming CH, the two conjectures together imply that all maximal Hardy fields are isomorphic.The proof we have in mind for the second conjecture depends on the first. Indeed, assumingthe first conjecture we can show that any countable pseudocauchy sequence in a Hardy fieldhas a pseudolimit in a Hardy field extension. This is one key step in the intended proof.Enough about Hardy fields for now. Let us turn to transseries.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20188/ 16

What are transseries?Also called logarithmic-exponential series, they are formal series in a variable x involvingtypically exp and log. One can get a sense by considering an example like:xee ex/2 ex/4 ···2 3 ex 5x 2 (log x)π 1 x 1 x 2 · · · e x .Think of x as positive infinite: x R. The monomials here, called transmonomials, arearranged from left to right in decreasing order, with real coefficients.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20189/ 16

What are transseries?Also called logarithmic-exponential series, they are formal series in a variable x involvingtypically exp and log. One can get a sense by considering an example like:xee ex/2 ex/4 ···2 3 ex 5x 2 (log x)π 1 x 1 x 2 · · · e x .Think of x as positive infinite: x R. The monomials here, called transmonomials, arearranged from left to right in decreasing order, with real coefficients.The field T of transseries has a somewhat lengthy inductive definition. For each transseriesthere is a finite bound on the “nesting” of exp and log in its transmonomials: series like1x 1ex 1e ex 1xe ee ··· ,1x 1x log x 1x log x log log x ···are excluded. (“T is not spherically complete.”)T is a real closed ordered field extension of R.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20189/ 16

T as a differential fieldEvery f T can be differentiated term by term:!0 Xex 1 n x n!xe.xn 0Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20181/ 16

T as a differential fieldEvery f T can be differentiated term by term:!0 Xex 1 n x n!xe.xn 0We obtain a derivation f 7 f 0 : T T on the field T:(f g ) f 0 g 0 ,(f · g )0 f 0 · g f · g 0 .Its constant field is {f T : f 0 0} R.Lou van den DriesHardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20181/ 16

T as a differential fieldEvery f T can be differentiated term by term:!0 Xex 1 n x n!xe.xn 0We obtain a derivation f 7 f 0 : T T on the field T:(f g ) f 0 g 0 ,(f · g )0 f 0 · g f · g 0 .Its constant field is {f T : f 0 0} R.Every f T has an antiderivative in T:Z x Xedx constant n!x 1 n exxn 0Lou van den DriesHardy Fields, Transseries, and Surreal Numbers(diverges).PAULO RIBENBOIM DAY at IHP, March 20, 20181/ 16

The dominance relation 4 on TFor f , g T,f 4g: f g: f 4 g and g 4 ff g: f 4 g and f 6 gFor example f 6 c g for some positive constant c0 e x x 10 1 log x x 1/10 ex eexAs in Hardy fields, f R f 0 0, and we can differentiate and integrate dominance:f 4 g f 0 4 g 0Lou van den Driesfor nonzero f , g 6 1.Hardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20181/ 16

T as an ordered valued differential fieldWe shall consider T as a valued ordered differential field, and model-theoretically as anL-structure where the language L has primitives0,1, , ,Lou van den Dries· , (derivation), 6 (ordering),4 (dominance).Hardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20181/ 16

T as an ordered valued differential fieldWe shall consider T as a valued ordered differential field, and model-theoretically as anL-structure where the language L has primitives0,1, , ,· , (derivation), 6 (ordering),4 (dominance).More generally, let K be any ordered differential field with constant fieldC {f K : f 0 0}. This yields a dominance relation 4 on K byf 4 g : f 6 c g for some positive c Cand we view K accordingly as an L-structure. We also introduce the valuation ring O of K ,O : {f K : f 4 1} convex hull of C in Kwith its maximal idealLou van den DriesO: {f K : f 1} of infinitesimals.Hardy Fields, Transseries, and Surreal NumbersPAULO RIBENBOIM DAY at IHP, March 20, 20181/ 16