The study of differential equations and the study of algebraic geometry are two disciplines within mathematics that seem to be mostly disjoint from each other. Looking deeper, however, one finds that connections do exist. This thesis gives in four chapters four examples of interesting mathematical insights that can be gained from combining the concepts and techniques from both of these fields. The first project shows how the behaviour of solutions of certain differential equations can be better understood by considering algebraic curves with a differential form. The second project proves the existence of certain higher differential operators in algebraic settings where these were not known to occur before. The third project shows that the existence or non-existence of power series solutions of partial differential equations can be interpreted from the perspective of tropical geometry. And the last project relates the old theorem of Siegel about integral points on elliptic curves to the monodromy of linear differential equations on this elliptic curve

Noordman, Marc Paul

English

ARTS repository - University of Groningen

   University of GroningenTopics in algebra, geometry and differential equationsNoordman, Marc PaulDOI:10.33612/diss.225415066IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2022Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordman, M. P. (2022). Topics in algebra, geometry and differential equations. [Thesis fully internal (DIV),University of Groningen]. University of Groningen. https://doi.org/10.33612/diss.225415066CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 10-09-2023Propositionsbelonging to the thesisTopics in algebra, geometry and dierentialequationsbyMarc Paul Noordman1. Dierential equations of the formP (u, u′) = 0, whereP ∈ C[X,Y ] is anirreducible polynomial involving both variables and C is an algebraicallyclosed eld of constants of characteristic 0, naturally fall into four classes.For the classes of exact, exponential type and Weierstrass type equations,algebraic relations between non-constant solutions are common, whilefor the class of general type equations, such relations are rare. Moreover,algebraic relations between non-constant solutions of dierent, pairwisenon-isogenous dierential equations of general type do not occur.2. e study of dierential equations of the form indicated in Proposition 1is greatly aided by tools from algebraic geometry – in particular, the the-ory of dierential forms and generalized Jacobian varieties aached tosmooth curves.3. Given a eld C of characteristic p > 0, a formal group law F ∈ C[[X,Y ]]of height ≥ 2, a separable extension K/C(t) and a C-linear derivationD : K → K such that Dp = 0, there exists an F -iterative derivation∂ : K → K[[T ]] such that ∂1 = D. Moreover, the components of such aderivation can be explicitly constructed.4. e Zassenhaus formula, which describes for non-commuting variablesA and B the expression (A + B)n in terms of iterated commutatorsadnA(B) (where adX(Y ) = [X,Y ] = XY − Y X), can be regarded as aspecial case, corresponding to the formal group law of the additive group,of a more general formula aached to any formal group law in charac-teristic 0.5. Let G ⊆ K[[t1, . . . , tm]]{x1, . . . , xn} be a dierential ideal, where Kis an uncountable algebraically closed eld of characteristic 0. e sup-port of any power series solution of G is constrained by the need forcancellations to occur. If for a potential support these constraints aresatised, then a power series solution with that support indeed exists.ese cancellation constraints can be equivalently formulated in termsof monomial-freeness of certain initial ideals. is leads to a “tropical”description of the set of supports of power series solutions of G.6. e tropical description of the set of supports of power series solutionsin Proposition 5 does not hold in general when K = Q, nor when onereplaces power series with Laurent series or Puiseux series.7. Siegel’s theorem on the niteness of the set of integral points on an ellip-tic curve can be reproven using the method of Lawrence and Venkatesh.For this purpose one can use an explicitly constructed nite-by-abeliancover of a punctured elliptic curve obtained from the Legendre family ofelliptic curves over P1 \ {0, 1,∞}.8. While proving theorems in the most general seing formally providesthe most powerful versions of these theorems, such generality comes ata practical cost when readers need to invest a large amount of time andeort in order to understand the theory. A researcher interested in pro-ducing results that are useful to the research community at large shouldcarefully weigh the advantages of a more general seing against this cost.Propositions 1 and 2 are based on joined work with Marius van der Put andJaap Top. Propositions 5 and 6 are based on joined work with François Boulier,Sebastian Falkensteiner, Cristhian Garay-López, Mercedes Haiech and ZeinabToghani.

Topics in algebra, geometry and differential equations

   University of GroningenTopics in algebra, geometry and differential equationsNoordman, Marc PaulDOI:10.33612/diss.225415066IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2022Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordman, M. P. (2022). Topics in algebra, geometry and differential equations. [Thesis fully internal (DIV),University of Groningen]. University of Groningen. https://doi.org/10.33612/diss.225415066CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 13-02-2023Propositionsbelonging to the thesisTopics in algebra, geometry and dierentialequationsbyMarc Paul Noordman1. Dierential equations of the formP (u, u′) = 0, whereP ∈ C[X,Y ] is anirreducible polynomial involving both variables and C is an algebraicallyclosed eld of constants of characteristic 0, naturally fall into four classes.For the classes of exact, exponential type and Weierstrass type equations,algebraic relations between non-constant solutions are common, whilefor the class of general type equations, such relations are rare. Moreover,algebraic relations between non-constant solutions of dierent, pairwisenon-isogenous dierential equations of general type do not occur.2. e study of dierential equations of the form indicated in Proposition 1is greatly aided by tools from algebraic geometry – in particular, the the-ory of dierential forms and generalized Jacobian varieties aached tosmooth curves.3. Given a eld C of characteristic p > 0, a formal group law F ∈ C[[X,Y ]]of height ≥ 2, a separable extension K/C(t) and a C-linear derivationD : K → K such that Dp = 0, there exists an F -iterative derivation∂ : K → K[[T ]] such that ∂1 = D. Moreover, the components of such aderivation can be explicitly constructed.4. e Zassenhaus formula, which describes for non-commuting variablesA and B the expression (A + B)n in terms of iterated commutatorsadnA(B) (where adX(Y ) = [X,Y ] = XY − Y X), can be regarded as aspecial case, corresponding to the formal group law of the additive group,of a more general formula aached to any formal group law in charac-teristic 0.5. Let G ⊆ K[[t1, . . . , tm]]{x1, . . . , xn} be a dierential ideal, where Kis an uncountable algebraically closed eld of characteristic 0. e sup-port of any power series solution of G is constrained by the need forcancellations to occur. If for a potential support these constraints aresatised, then a power series solution with that support indeed exists.ese cancellation constraints can be equivalently formulated in termsof monomial-freeness of certain initial ideals. is leads to a “tropical”description of the set of supports of power series solutions of G.6. e tropical description of the set of supports of power series solutionsin Proposition 5 does not hold in general when K = Q, nor when onereplaces power series with Laurent series or Puiseux series.7. Siegel’s theorem on the niteness of the set of integral points on an ellip-tic curve can be reproven using the method of Lawrence and Venkatesh.For this purpose one can use an explicitly constructed nite-by-abeliancover of a punctured elliptic curve obtained from the Legendre family ofelliptic curves over P1 \ {0, 1,∞}.8. While proving theorems in the most general seing formally providesthe most powerful versions of these theorems, such generality comes ata practical cost when readers need to invest a large amount of time andeort in order to understand the theory. A researcher interested in pro-ducing results that are useful to the research community at large shouldcarefully weigh the advantages of a more general seing against this cost.Propositions 1 and 2 are based on joined work with Marius van der Put andJaap Top. Propositions 5 and 6 are based on joined work with François Boulier,Sebastian Falkensteiner, Cristhian Garay-López, Mercedes Haiech and ZeinabToghani.

University of Groningen

   University of GroningenTopics in algebra, geometry and differential equationsNoordman, Marc PaulDOI:10.33612/diss.225415066IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2022Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordman, M. P. (2022). Topics in algebra, geometry and differential equations. University of Groningen.https://doi.org/10.33612/diss.225415066CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 20-11-2022Propositionsbelonging to the thesisTopics in algebra, geometry and dierentialequationsbyMarc Paul Noordman1. Dierential equations of the formP (u, u′) = 0, whereP ∈ C[X,Y ] is anirreducible polynomial involving both variables and C is an algebraicallyclosed eld of constants of characteristic 0, naturally fall into four classes.For the classes of exact, exponential type and Weierstrass type equations,algebraic relations between non-constant solutions are common, whilefor the class of general type equations, such relations are rare. Moreover,algebraic relations between non-constant solutions of dierent, pairwisenon-isogenous dierential equations of general type do not occur.2. e study of dierential equations of the form indicated in Proposition 1is greatly aided by tools from algebraic geometry – in particular, the the-ory of dierential forms and generalized Jacobian varieties aached tosmooth curves.3. Given a eld C of characteristic p > 0, a formal group law F ∈ C[[X,Y ]]of height ≥ 2, a separable extension K/C(t) and a C-linear derivationD : K → K such that Dp = 0, there exists an F -iterative derivation∂ : K → K[[T ]] such that ∂1 = D. Moreover, the components of such aderivation can be explicitly constructed.4. e Zassenhaus formula, which describes for non-commuting variablesA and B the expression (A + B)n in terms of iterated commutatorsadnA(B) (where adX(Y ) = [X,Y ] = XY − Y X), can be regarded as aspecial case, corresponding to the formal group law of the additive group,of a more general formula aached to any formal group law in charac-teristic 0.5. Let G ⊆ K[[t1, . . . , tm]]{x1, . . . , xn} be a dierential ideal, where Kis an uncountable algebraically closed eld of characteristic 0. e sup-port of any power series solution of G is constrained by the need forcancellations to occur. If for a potential support these constraints aresatised, then a power series solution with that support indeed exists.ese cancellation constraints can be equivalently formulated in termsof monomial-freeness of certain initial ideals. is leads to a “tropical”description of the set of supports of power series solutions of G.6. e tropical description of the set of supports of power series solutionsin Proposition 5 does not hold in general when K = Q, nor when onereplaces power series with Laurent series or Puiseux series.7. Siegel’s theorem on the niteness of the set of integral points on an ellip-tic curve can be reproven using the method of Lawrence and Venkatesh.For this purpose one can use an explicitly constructed nite-by-abeliancover of a punctured elliptic curve obtained from the Legendre family ofelliptic curves over P1 \ {0, 1,∞}.8. While proving theorems in the most general seing formally providesthe most powerful versions of these theorems, such generality comes ata practical cost when readers need to invest a large amount of time andeort in order to understand the theory. A researcher interested in pro-ducing results that are useful to the research community at large shouldcarefully weigh the advantages of a more general seing against this cost.Propositions 1 and 2 are based on joined work with Marius van der Put andJaap Top. Propositions 5 and 6 are based on joined work with François Boulier,Sebastian Falkensteiner, Cristhian Garay-López, Mercedes Haiech and ZeinabToghani.

Dissertations of the University of Groningen

   University of GroningenTopics in algebra, geometry and differential equationsNoordman, Marc PaulDOI:10.33612/diss.225415066IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2022Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordman, M. P. (2022). Topics in algebra, geometry and differential equations. [Thesis fully internal (DIV),University of Groningen]. University of Groningen. https://doi.org/10.33612/diss.225415066CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 31-10-2023Propositionsbelonging to the thesisTopics in algebra, geometry and dierentialequationsbyMarc Paul Noordman1. Dierential equations of the formP (u, u′) = 0, whereP ∈ C[X,Y ] is anirreducible polynomial involving both variables and C is an algebraicallyclosed eld of constants of characteristic 0, naturally fall into four classes.For the classes of exact, exponential type and Weierstrass type equations,algebraic relations between non-constant solutions are common, whilefor the class of general type equations, such relations are rare. Moreover,algebraic relations between non-constant solutions of dierent, pairwisenon-isogenous dierential equations of general type do not occur.2. e study of dierential equations of the form indicated in Proposition 1is greatly aided by tools from algebraic geometry – in particular, the the-ory of dierential forms and generalized Jacobian varieties aached tosmooth curves.3. Given a eld C of characteristic p > 0, a formal group law F ∈ C[[X,Y ]]of height ≥ 2, a separable extension K/C(t) and a C-linear derivationD : K → K such that Dp = 0, there exists an F -iterative derivation∂ : K → K[[T ]] such that ∂1 = D. Moreover, the components of such aderivation can be explicitly constructed.4. e Zassenhaus formula, which describes for non-commuting variablesA and B the expression (A + B)n in terms of iterated commutatorsadnA(B) (where adX(Y ) = [X,Y ] = XY − Y X), can be regarded as aspecial case, corresponding to the formal group law of the additive group,of a more general formula aached to any formal group law in charac-teristic 0.5. Let G ⊆ K[[t1, . . . , tm]]{x1, . . . , xn} be a dierential ideal, where Kis an uncountable algebraically closed eld of characteristic 0. e sup-port of any power series solution of G is constrained by the need forcancellations to occur. If for a potential support these constraints aresatised, then a power series solution with that support indeed exists.ese cancellation constraints can be equivalently formulated in termsof monomial-freeness of certain initial ideals. is leads to a “tropical”description of the set of supports of power series solutions of G.6. e tropical description of the set of supports of power series solutionsin Proposition 5 does not hold in general when K = Q, nor when onereplaces power series with Laurent series or Puiseux series.7. Siegel’s theorem on the niteness of the set of integral points on an ellip-tic curve can be reproven using the method of Lawrence and Venkatesh.For this purpose one can use an explicitly constructed nite-by-abeliancover of a punctured elliptic curve obtained from the Legendre family ofelliptic curves over P1 \ {0, 1,∞}.8. While proving theorems in the most general seing formally providesthe most powerful versions of these theorems, such generality comes ata practical cost when readers need to invest a large amount of time andeort in order to understand the theory. A researcher interested in pro-ducing results that are useful to the research community at large shouldcarefully weigh the advantages of a more general seing against this cost.Propositions 1 and 2 are based on joined work with Marius van der Put andJaap Top. Propositions 5 and 6 are based on joined work with François Boulier,Sebastian Falkensteiner, Cristhian Garay-López, Mercedes Haiech and ZeinabToghani.

   University of GroningenTopics in algebra, geometry and differential equationsNoordman, Marc PaulDOI:10.33612/diss.225415066IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2022Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordman, M. P. (2022). Topics in algebra, geometry and differential equations. [Thesis fully internal (DIV),University of Groningen]. University of Groningen. https://doi.org/10.33612/diss.225415066CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-04-2023Propositionsbelonging to the thesisTopics in algebra, geometry and dierentialequationsbyMarc Paul Noordman1. Dierential equations of the formP (u, u′) = 0, whereP ∈ C[X,Y ] is anirreducible polynomial involving both variables and C is an algebraicallyclosed eld of constants of characteristic 0, naturally fall into four classes.For the classes of exact, exponential type and Weierstrass type equations,algebraic relations between non-constant solutions are common, whilefor the class of general type equations, such relations are rare. Moreover,algebraic relations between non-constant solutions of dierent, pairwisenon-isogenous dierential equations of general type do not occur.2. e study of dierential equations of the form indicated in Proposition 1is greatly aided by tools from algebraic geometry – in particular, the the-ory of dierential forms and generalized Jacobian varieties aached tosmooth curves.3. Given a eld C of characteristic p > 0, a formal group law F ∈ C[[X,Y ]]of height ≥ 2, a separable extension K/C(t) and a C-linear derivationD : K → K such that Dp = 0, there exists an F -iterative derivation∂ : K → K[[T ]] such that ∂1 = D. Moreover, the components of such aderivation can be explicitly constructed.4. e Zassenhaus formula, which describes for non-commuting variablesA and B the expression (A + B)n in terms of iterated commutatorsadnA(B) (where adX(Y ) = [X,Y ] = XY − Y X), can be regarded as aspecial case, corresponding to the formal group law of the additive group,of a more general formula aached to any formal group law in charac-teristic 0.5. Let G ⊆ K[[t1, . . . , tm]]{x1, . . . , xn} be a dierential ideal, where Kis an uncountable algebraically closed eld of characteristic 0. e sup-port of any power series solution of G is constrained by the need forcancellations to occur. If for a potential support these constraints aresatised, then a power series solution with that support indeed exists.ese cancellation constraints can be equivalently formulated in termsof monomial-freeness of certain initial ideals. is leads to a “tropical”description of the set of supports of power series solutions of G.6. e tropical description of the set of supports of power series solutionsin Proposition 5 does not hold in general when K = Q, nor when onereplaces power series with Laurent series or Puiseux series.7. Siegel’s theorem on the niteness of the set of integral points on an ellip-tic curve can be reproven using the method of Lawrence and Venkatesh.For this purpose one can use an explicitly constructed nite-by-abeliancover of a punctured elliptic curve obtained from the Legendre family ofelliptic curves over P1 \ {0, 1,∞}.8. While proving theorems in the most general seing formally providesthe most powerful versions of these theorems, such generality comes ata practical cost when readers need to invest a large amount of time andeort in order to understand the theory. A researcher interested in pro-ducing results that are useful to the research community at large shouldcarefully weigh the advantages of a more general seing against this cost.Propositions 1 and 2 are based on joined work with Marius van der Put andJaap Top. Propositions 5 and 6 are based on joined work with François Boulier,Sebastian Falkensteiner, Cristhian Garay-López, Mercedes Haiech and ZeinabToghani.

Proceedings - University of Groningen

   University of GroningenTopics in algebra, geometry and differential equationsNoordman, Marc PaulDOI:10.33612/diss.225415066IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2022Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordman, M. P. (2022). Topics in algebra, geometry and differential equations. University of Groningen.https://doi.org/10.33612/diss.225415066CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 29-10-2022Propositionsbelonging to the thesisTopics in algebra, geometry and dierentialequationsbyMarc Paul Noordman1. Dierential equations of the formP (u, u′) = 0, whereP ∈ C[X,Y ] is anirreducible polynomial involving both variables and C is an algebraicallyclosed eld of constants of characteristic 0, naturally fall into four classes.For the classes of exact, exponential type and Weierstrass type equations,algebraic relations between non-constant solutions are common, whilefor the class of general type equations, such relations are rare. Moreover,algebraic relations between non-constant solutions of dierent, pairwisenon-isogenous dierential equations of general type do not occur.2. e study of dierential equations of the form indicated in Proposition 1is greatly aided by tools from algebraic geometry – in particular, the the-ory of dierential forms and generalized Jacobian varieties aached tosmooth curves.3. Given a eld C of characteristic p > 0, a formal group law F ∈ C[[X,Y ]]of height ≥ 2, a separable extension K/C(t) and a C-linear derivationD : K → K such that Dp = 0, there exists an F -iterative derivation∂ : K → K[[T ]] such that ∂1 = D. Moreover, the components of such aderivation can be explicitly constructed.4. e Zassenhaus formula, which describes for non-commuting variablesA and B the expression (A + B)n in terms of iterated commutatorsadnA(B) (where adX(Y ) = [X,Y ] = XY − Y X), can be regarded as aspecial case, corresponding to the formal group law of the additive group,of a more general formula aached to any formal group law in charac-teristic 0.5. Let G ⊆ K[[t1, . . . , tm]]{x1, . . . , xn} be a dierential ideal, where Kis an uncountable algebraically closed eld of characteristic 0. e sup-port of any power series solution of G is constrained by the need forcancellations to occur. If for a potential support these constraints aresatised, then a power series solution with that support indeed exists.ese cancellation constraints can be equivalently formulated in termsof monomial-freeness of certain initial ideals. is leads to a “tropical”description of the set of supports of power series solutions of G.6. e tropical description of the set of supports of power series solutionsin Proposition 5 does not hold in general when K = Q, nor when onereplaces power series with Laurent series or Puiseux series.7. Siegel’s theorem on the niteness of the set of integral points on an ellip-tic curve can be reproven using the method of Lawrence and Venkatesh.For this purpose one can use an explicitly constructed nite-by-abeliancover of a punctured elliptic curve obtained from the Legendre family ofelliptic curves over P1 \ {0, 1,∞}.8. While proving theorems in the most general seing formally providesthe most powerful versions of these theorems, such generality comes ata practical cost when readers need to invest a large amount of time andeort in order to understand the theory. A researcher interested in pro-ducing results that are useful to the research community at large shouldcarefully weigh the advantages of a more general seing against this cost.Propositions 1 and 2 are based on joined work with Marius van der Put andJaap Top. Propositions 5 and 6 are based on joined work with François Boulier,Sebastian Falkensteiner, Cristhian Garay-López, Mercedes Haiech and ZeinabToghani.

   University of GroningenTopics in algebra, geometry and differential equationsNoordman, Marc PaulDOI:10.33612/diss.225415066IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2022Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordman, M. P. (2022). Topics in algebra, geometry and differential equations. [Thesis fully internal (DIV),University of Groningen]. University of Groningen. https://doi.org/10.33612/diss.225415066CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 28-12-2022Propositionsbelonging to the thesisTopics in algebra, geometry and dierentialequationsbyMarc Paul Noordman1. Dierential equations of the formP (u, u′) = 0, whereP ∈ C[X,Y ] is anirreducible polynomial involving both variables and C is an algebraicallyclosed eld of constants of characteristic 0, naturally fall into four classes.For the classes of exact, exponential type and Weierstrass type equations,algebraic relations between non-constant solutions are common, whilefor the class of general type equations, such relations are rare. Moreover,algebraic relations between non-constant solutions of dierent, pairwisenon-isogenous dierential equations of general type do not occur.2. e study of dierential equations of the form indicated in Proposition 1is greatly aided by tools from algebraic geometry – in particular, the the-ory of dierential forms and generalized Jacobian varieties aached tosmooth curves.3. Given a eld C of characteristic p > 0, a formal group law F ∈ C[[X,Y ]]of height ≥ 2, a separable extension K/C(t) and a C-linear derivationD : K → K such that Dp = 0, there exists an F -iterative derivation∂ : K → K[[T ]] such that ∂1 = D. Moreover, the components of such aderivation can be explicitly constructed.4. e Zassenhaus formula, which describes for non-commuting variablesA and B the expression (A + B)n in terms of iterated commutatorsadnA(B) (where adX(Y ) = [X,Y ] = XY − Y X), can be regarded as aspecial case, corresponding to the formal group law of the additive group,of a more general formula aached to any formal group law in charac-teristic 0.5. Let G ⊆ K[[t1, . . . , tm]]{x1, . . . , xn} be a dierential ideal, where Kis an uncountable algebraically closed eld of characteristic 0. e sup-port of any power series solution of G is constrained by the need forcancellations to occur. If for a potential support these constraints aresatised, then a power series solution with that support indeed exists.ese cancellation constraints can be equivalently formulated in termsof monomial-freeness of certain initial ideals. is leads to a “tropical”description of the set of supports of power series solutions of G.6. e tropical description of the set of supports of power series solutionsin Proposition 5 does not hold in general when K = Q, nor when onereplaces power series with Laurent series or Puiseux series.7. Siegel’s theorem on the niteness of the set of integral points on an ellip-tic curve can be reproven using the method of Lawrence and Venkatesh.For this purpose one can use an explicitly constructed nite-by-abeliancover of a punctured elliptic curve obtained from the Legendre family ofelliptic curves over P1 \ {0, 1,∞}.8. While proving theorems in the most general seing formally providesthe most powerful versions of these theorems, such generality comes ata practical cost when readers need to invest a large amount of time andeort in order to understand the theory. A researcher interested in pro-ducing results that are useful to the research community at large shouldcarefully weigh the advantages of a more general seing against this cost.Propositions 1 and 2 are based on joined work with Marius van der Put andJaap Top. Propositions 5 and 6 are based on joined work with François Boulier,Sebastian Falkensteiner, Cristhian Garay-López, Mercedes Haiech and ZeinabToghani.

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Topics in algebra, geometry and differential equations

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