> <\body> A differential intermediate value theorem >>> \; \; \; <\with|paragraph mode|center> \\>> \; \; C.N.R.S., Université Paris-Sud \; \; \\>> \; \; \; ||>|>>>acs>>> ()>>>>>>> > Field > of grid-based transseries in 1>: <\itemize> Serial structure and total relation >. Stable under exp and log. Real algebraically closed. Stable under differentiation, integration. Stable under composition, inversion. <\description> <\itemize> Dahn & Göring, 1984, 1986 Van den Dries, Ressayre, etc. 199* Écalle, 1989, 1992> <\itemize> Shackell, Salvy, etc. 1990-* vdH, 1994-* <\itemize> +x+x+x+x+\> +>+\+e+|x>+\+e+\> +3+4+\> +x>+x>+x>+\> +>+>>+\> +|x>+|x>+\>+x*e+|x>+|x>+\>+\> (x-\)+log \(e(x)>)*x>>> +>+>>+>>>+\> +f(x)+f(e x>)> +e+e+\>>>> <\itemize> Totally ordered constant field . Monomial group >, with total ordering >. [Hahn 1907] Set of ]]={f:\\C\|supp f}> forms a totally ordered field. *\*(1+\)> g\\\\> Canonical decomposition: >>|+>|>>>|>|>|>>|\1>f>*\>|>|\1>f>*\>>>>> grid-based >> ,\,\\1> and ,\,\> : {\,\,\}>*{\,\,\}.> >\C[[\]]>: field of . >> Start with monomial group =\=x>*(log x)>*(log log x)>*\> and logarithm on >>>: >*\log> x(1+\))=>>|*log x+\+\*log x+log (1+\).>>>>>> Assume > given, with logarithm on >>>. =exp \>>,> with \>\exp g>\f\g.> Take \>>*(1+\))=log c+f\>+log (1+\).> Inductive limit: =C\\\\>>. > (1++>+\)>\\>. A is a tuple =(\,\,\)> with <\description> \\\\\> transmonomials. =exp x>, with \>. \exp \;\;\>> for 1>. Recursive expansion of C;\;\>>: ||>f>*\>;>>|>>||>f,\>*\>;>>||>|>|,\,\>>||>f,\,\>*\>.>>>>> > be a transbasis and \> a transseries. Then there exists a supertransbasis \\>, such that \>.>> > |1-x>>)\\+x>>>. >> For =x>*\(log x)>\\>: =\*|x>+\+|x*\*log x>.> ``Extension by strong linearity'' to =\>>. >> For =e\>>\exp \>>>: =(f\>)*\> ``Extension by strong linearity'' to >>. <\itemize> =f\exp>. =f\log.> > ||P>>|>||+\+i=d>P,\,i>*fi>*\*(f)>>>|||+\+i\|P,\,i>\0}>>|P\<\|\|\>>||+\+r*i\|P,\,i>\0}>>|(f)>||>|h>(f)>||>|\(f\)>||>>>>> >||> {\,\,i>>\|P,\,i>\0}>>|>||,\,i>P,\,i,\>*c>*\*(c)>>>>>> <\theorem> For some \<\|\|\>P\<\|\|\>>, there exists a =N\\[c]*(c)>>, such that for every k>: >=N.> > ||-(f)>>|\>||*(f*f-f*f-(f))>>|\\>||-x>*f*f+e-2*x>*(f*f-(f))>>|>||>>>>> > : Newton polynomial associated to . P(f)=0(f\\)> <\itemize> \\> > dom.> monomial> if \>>\C>. =c*\\\> > dom.> term> if \>>(c)=0.> of >: mult. of as root of P\>>>. : max.> deg.> of P\>>> (\\>). <\description> \>>\\[c]\\C>. \>>\\*(c)>>>. \>>\(\[c]\\\)*(c)>>.> \0> and \0>. Then there exists a unique equalizer =\>, such that +P)\>>> is non homogeneous.> Ricatti polynomials: (f)=R(f>)*fi>> <\proposition> The monomial \\> is a potential dominant monomial w.r.t. (f)=0> if and only if >>(f>)=0(f>\>)>> has strictly positive Newton degree. Change of variables + asymptotic constraint f=\+(\|~>)> transforms () into P>()=0(\|~>)> > be the dominant term of >. Then the Newton degree of )> equals the multiplicity of > as a potential dominant term for )>.> : of Newton degree 1. Newton degree does not decrease for refinement +(\|~>)> > consider refinements like +(\|~>),> with P|\f> (\)=0(\\\).> > unravelling: refinement () with <\description> The Newton degree of () is . For any |~>\|~>>, the Newton degree of +|~>>(|~>)=0(|~>\\(|~>))> is d>. are all purely exponential, then there exists an integer >, such that the logarithmic depth of each root of is bounded by >.> -(f).> Algebraic potential dominant term: =>*x.> Differential potential dominant terms: *e*x>> with \0>. Further terms of expansion... > be a differential polynomial with coefficients in >. If \\> are such that )*P(\)\0>, then there exists an (\,\)>, with .> Example: +e>*f3>*f+\(log \(x)+1)=0.> Generalize the theorem to include > and > like <\itemize> =\\\>>. =\\\>>. =\\\*\> =\\\*\.> =\\\>, where =>>. >>f)> has constant sign for sufficiently large \>.> in >,f\>,\>> >>\)> has constant sign for sufficiently small \>.> <\itemize> (f)=sign P(f)>. g\log \|f\|\log \|g\|>. =Q(c)*(c)>> and purely exponential. > multiplicity of as a root of . <\lemma> For all \\1> with \e>, the signs of )> and )> are independent of > and given by >)>*\(c-\)=(>)>*\(c+\)=\)>>(c).> <\lemma> For homogeneous of degree : (\)=\(\)=\(\>)>=\(-\)>> for all \\1> with \e>. <\lemma> For all f\1> with e>, the signs of >)> and are independent of and given by >)>*\(-\)=\(\)=sign Q.> <\lemma> For homogeneous of degree : (\)=\(f)=\>(f>)=\(\)>,> for all f\1> with e>. <\itemize> Reduction to intervals of the form ,\+\*\)> or +\,\+\*\)>. Triple induction over order of , Newton degree of () and maximal length of chain of ``privileged refinements''. Reduce interval using lemmas and by looking where the sign change occurs. > <\itemize> Complex transseries <\expand|itemize-minus> At least complex transseries solutions. Linear differential operators factorize. Case separations & corresponding regions. Analysable functions. Differential-difference equations <\expand|itemize-minus> OK for postcompositions with exponentiality (like )>. Transseries of positive strength (PhD. Michael Schmeling). Nested transseries. Model theory. Algorithms. And much and much more... \; <\initial> <\collection> <\references> <\collection> > |6>> > > |6>> > > > > > > > > > > > > > > > > > > > > > > > > > |2>> |2>> |2>> > |3>> |4>> |4>> |5>> |5>> |5>> <\auxiliary> <\collection> <\associate|toc> PART I Transseries Origins Examples Well-ordered power series Grid-based series Construction of |\> |Logarithmic transseries> |Inductive step> Transbases |Incomplete transbasis theorem> Differentiation |Differentiation on |\>> |Differentiation on |\>> Upward & downward shifting Part II |Algebraic differential polynomials> Dominant parts Differential Newton polygons |Newton degree> |Potential dominant monomial types> |Algebraic and mixed p.d.m.s> |Differential p.d.m.s> |Refinements> |Quasi-linear equations> |Unravellings> |Structure of solutions> |Example> |Exercise> Part III |Intermediate value theorem> |Proof strategy> |Behaviour near zero and infinity> |Behaviour near constants> |Behaviour before & after constants> |Final proof> What next?