Piotr Kowalski

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All published works (36)

Positive characteristic Ax–Schanuel

2024-07-19

Piotr Kowalski

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The class of Krasner hyperfields is not elementary

2024-04-22

Piotr Błaszkiewicz , Piotr Kowalski

We show that the class of Krasner hyperfields is not elementary. To show this, we determine the rational rank of quotients of multiplicative groups in field extensions. Our argument uses … We show that the class of Krasner hyperfields is not elementary. To show this, we determine the rational rank of quotients of multiplicative groups in field extensions. Our argument uses Chebotarev's density theorem. We also discuss some related questions.

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Corrigendum: Model theory of fields with virtually free group actions

2024-04-01

Özlem Beyarslan , Piotr Kowalski

Abstract There is an irreparable error in the proof of Theorem 3.26 in the above‐mentioned paper and we withdraw the claim of having proved that theorem. In fact, that theorem … Abstract There is an irreparable error in the proof of Theorem 3.26 in the above‐mentioned paper and we withdraw the claim of having proved that theorem. In fact, that theorem is false in a very strong sense.

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PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS – ERRATUM

2024-03-26

Daniel Max Hoffmann , Piotr Kowalski

An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also … An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also available in through the 'Save PDF' action button.

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PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS

2023-10-20

Daniel Max Hoffmann , Piotr Kowalski

Abstract We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of … Abstract We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property is first order.

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Of model completeness and algebraic groups

2023-01-01

Daniel Max Hoffmann , Piotr Kowalski , Chieu-Minh Tran +1 more

We show that if G is a simply connected semi-simple algebraic group and K is a model complete field, then the theory of the group G(K) is model complete as … We show that if G is a simply connected semi-simple algebraic group and K is a model complete field, then the theory of the group G(K) is model complete as well.

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Difference sheaves and torsors

2022-10-27

Marcin Chałupnik , Piotr Kowalski

We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, … We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, difference Picard group and a

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MODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS

2022-06-30

Özlem Beyarslan , Piotr Kowalski

Abstract We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p , the p -primary part … Abstract We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p , the p -primary part of A is either finite or it coincides with the Prüfer p -group. We also provide a model-theoretic description of the model companions we obtain.

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Operators coming from ring schemes

2022-05-03

Jakub Gogolok , Piotr Kowalski

We introduce the notion of a coordinate k ${\bf k}$ -algebra scheme and the corresponding notion of a B $\mathcal {B}$ -operator. This class of operators includes endomorphisms and derivations … We introduce the notion of a coordinate k ${\bf k}$ -algebra scheme and the corresponding notion of a B $\mathcal {B}$ -operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it generalizes the operators related to D $\mathcal {D}$ -rings from Moosa and Scanlon (J. Math. Log. 14 (2014), no. 02, 1450009) as well. We classify the (coordinate) k ${\bf k}$ -algebra schemes for a perfect field k ${\bf k}$ and we also discuss the model-theoretic properties of fields with B $\mathcal {B}$ -operators.

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MODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS

2022-03-14

Daniel Max Hoffmann , Piotr Kowalski

Abstract We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such … Abstract We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

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Model theory of fields with free operators in positive characteristic

2019-06-05

Özlem Beyarslan , Daniel Max Hoffmann , Moshe Kamensky +1 more

We give algebraic conditions for a finite commutative algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a field of positive characteristic, which are … We give algebraic conditions for a finite commutative algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a field of positive characteristic, which are equivalent to the companionability of the theory of fields with “<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-operators” (i.e., the operators coming from homomorphisms into tensor products with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). We show that, in the most interesting case of a local <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, these model companions admit quantifier elimination in the “smallest possible” language, and they are strictly stable. We also describe the forking relation there.

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Difference modules and difference cohomology

2018-10-27

Marcin Chałupnik , Piotr Kowalski

We give some basics about homological algebra of difference representations. We consider both the difference-discrete and the difference-rational case. We define the corresponding cohomology theories and show the existence of … We give some basics about homological algebra of difference representations. We consider both the difference-discrete and the difference-rational case. We define the corresponding cohomology theories and show the existence of spectral sequences relating these cohomology theories with the standard ones.

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Model theory of fields with virtually free group actions

2018-07-26

Özlem Beyarslan , Piotr Kowalski

For a group G, we define the notion of a G-kernel and show that the properties of G-kernels are closely related with the existence of a model companion of the … For a group G, we define the notion of a G-kernel and show that the properties of G-kernels are closely related with the existence of a model companion of the theory of Galois actions of G. Using Bass–Serre theory, we show that this model companion exists for virtually free groups generalizing the existing results about free groups and finite groups. We show that the new theories we obtain are not simple and not even NTP 2 .

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Model theory of fields with free operators in positive characteristic

2018-06-01

Özlem Beyarslan , Daniel Max Hoffmann , Moshe Kamensky +1 more

We give algebraic conditions about a finite algebra $B$ over a perfect field of positive characteristic, which are equivalent to the companionability of the theory of fields with $B$-operators (i.e. … We give algebraic conditions about a finite algebra $B$ over a perfect field of positive characteristic, which are equivalent to the companionability of the theory of fields with $B$-operators (i.e. the operators coming from homomorphisms into tensor products with $B$). We show that, in the most interesting case of a local $B$, these model companions admit quantifier elimination in the smallest possible language and they are strictly stable. We also describe the forking relation there.

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Existentially closed fields with finite group actions

2017-11-24

Daniel Max Hoffmann , Piotr Kowalski

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong … We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action.

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AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC

2017-11-08

Piotr Kowalski

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic … We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

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STRONGLY MINIMAL REDUCTS OF VALUED FIELDS

2016-06-01

Piotr Kowalski , Serge Randriambololona

Abstract We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying … Abstract We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

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Existentially closed fields with<i>G</i>-derivations

2016-04-13

Daniel Max Hoffmann , Piotr Kowalski

We prove that the theories of fields with Hasse-Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions. We prove that the theories of fields with Hasse-Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions.

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A note on extending actions of infinitesimal group schemes

2015-11-16

Daniel Max Hoffmann , Piotr Kowalski

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Strongly minimal reducts of valued fields

2014-08-14

Piotr Kowalski , Serge Randriambololona

We prove that if a strongly minimal non-locally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field. We prove that if a strongly minimal non-locally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

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Integrating Hasse–Schmidt derivations

2014-05-16

Daniel Max Hoffmann , Piotr Kowalski

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Ax-Schanuel condition in arbitrary characteristic

2014-02-09

Piotr Kowalski

We prove a positive characteristic version of Ax's theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group. Our result is stated in a … We prove a positive characteristic version of Ax's theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax-Schanuel type.

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Schanuel property for additive power series

2011-10-06

Piotr Kowalski

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On the isotriviality of projective iterative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>∂</mml:mi></mml:math>-varieties

2011-06-28

Piotr Kowalski , Anand Pillay

A note on a theorem of Ax

2008-07-08

Piotr Kowalski

Strongly minimal expansions of (ℂ, +) definable in o-minimal fields

2008-01-04

Assaf Hasson , Piotr Kowalski

We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly … We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly after conjugating by a real matrix. In particular we prove a special case of the Zilber Dichotomy: an algebraically closed field is definable in certain strongly minimal structures which are definable in an o-minimal field.

On algebraic 𝜎-groups

2006-10-17

Piotr Kowalski , Anand Pillay

We introduce the categories of algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-varieties and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> … We introduce the categories of algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-varieties and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-groups over a difference field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper K comma sigma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(K,\sigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Under a “linearly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-closed" assumption on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper K comma sigma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(K,\sigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove an isotriviality theorem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-groups. This theorem immediately yields the key lemma in a proof of the Manin-Mumford conjecture. The present paper crucially uses ideas of Pilay and Ziegler (2003) but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski’s work (2001) and are also closely related to papers of Pink and Roessler (2002 and 2004).

Geometric axioms for existentially closed Hasse fields

2005-05-18

Piotr Kowalski

Jet operators on fields

2005-03-11

Piotr Kowalski

Derivations of the Frobenius map

2005-03-01

Piotr Kowalski

Abstract We prove that the theory of fields with a derivation of Frobenius has the model companion which is stable and admits elimination of quantifiers up to the level of … Abstract We prove that the theory of fields with a derivation of Frobenius has the model companion which is stable and admits elimination of quantifiers up to the level of the λ-functions. Along the way, we give new geometric axioms of DCF p .

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Quantifier elimination for algebraic $D$-groups

2005-01-21

Piotr Kowalski , Anand Pillay

We prove that if $G$ is an algebraic $D$-group (in the sense of Buium over a differentially closed field $(K,\partial )$ of characteristic $0$, then the first order structure consisting … We prove that if $G$ is an algebraic $D$-group (in the sense of Buium over a differentially closed field $(K,\partial )$ of characteristic $0$, then the first order structure consisting of $G$ together with the algebraic $D$-subvarieties of $G, G\times G,\dots$, has quantifier-elimination. In other words, the projection on $G^{n}$ of a $D$-constructible subset of $G^{n+1}$ is $D$-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.

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Lazard’s Theorem for differential algebraic groups and proalgebraic groups

2002-02-01

Marcin Chałupnik , Piotr Kowalski

We prove that a differential group whose underlying variety is an affine space is unipotent.The problem is reduced to an infinite-dimensional version of Lazard's Theorem. We prove that a differential group whose underlying variety is an affine space is unipotent.The problem is reduced to an infinite-dimensional version of Lazard's Theorem.

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A note on groups definable in difference fields

2001-05-22

Piotr Kowalski , Anthony L. Pillay

We prove that a group definable in a model of $ACFA$ is virtually definably embeddable in an algebraic group. We give an improved proof of the same result for groups … We prove that a group definable in a model of $ACFA$ is virtually definably embeddable in an algebraic group. We give an improved proof of the same result for groups definable in differentially closed fields. We also extend to the difference field context results on the unipotence of definable groups on affine spaces.

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Stable Groups and Algebraic Groups

2000-02-01

Piotr Kowalski

A certain property of some type-definable subgroups of superstable groups with finite U-rank is closely related to the Mordell–Lang conjecture. This property is discussed in the context of algebraic groups. A certain property of some type-definable subgroups of superstable groups with finite U-rank is closely related to the Mordell–Lang conjecture. This property is discussed in the context of algebraic groups.

Pro-algebraic and differential algebraic group structures on affine spaces

2000-02-01

Piotr Kowalski , Anand Pillay

We prove the following result, solving a problem raised in an article by A. Buium and Ph. J. Cassidy, to appear in the collected works of E. Kolchin. If G … We prove the following result, solving a problem raised in an article by A. Buium and Ph. J. Cassidy, to appear in the collected works of E. Kolchin. If G is a differential algebraic group whose underlying set is some affine n -space, then G is unipotent. A key result, possibly of independent interest, concerns infinite-dimensional group schemes: a group scheme whose underlying scheme is Spec ( K [ X 1 , X 2 ,...]) ( K an algebraically closed field of characteristic 0) is a projective limit of unipotent algebraic groups.

Limit Theorems for Maximal Random Sums

1998-01-01

Piotr Kowalski , Z. Rychlik

Common Coauthors

Coauthor	Papers Together
Daniel Max Hoffmann	10
Özlem Beyarslan	5
Anand Pillay	4
Marcin Chałupnik	3
Serge Randriambololona	2
Moshe Kamensky	2
Assaf Hasson	1
Z. Rychlik	1
Jakub Gogolok	1
Anthony L. Pillay	1
Piotr Błaszkiewicz	1
Chieu-Minh Tran	1
Jinhe Ye	1

Commonly Cited References

Model theory of difference fields

1999-04-08

Zoé Chatzidakis , Ehud Hrushovski

A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory … A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"><mml:semantics><mml:mn>0</mml:mn><mml:annotation encoding="application/x-tex">0</mml:annotation></mml:semantics></mml:math></inline-formula>.

Separably closed fields with Hasse derivations

2003-03-01

Martin Ziegler

Abstract In [6] Messmer and Wood proved quantifier elimination for separably closed fields of finite Ershov invariant e equipped with a (certain) Hasse derivation. We propose a variant of their … Abstract In [6] Messmer and Wood proved quantifier elimination for separably closed fields of finite Ershov invariant e equipped with a (certain) Hasse derivation. We propose a variant of their theory, using a sequence of e commuting Hasse derivations. In contrast to [6] our Hasse derivations are iterative.

Introduction to Affine Group Schemes

1979-01-01

William C. Waterhouse

Model theory of fields with free operators in characteristic zero

2014-09-04

Rahim Moosa , Thomas Scanlon

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme 𝒟 over a field A of characteristic zero, the … Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme 𝒟 over a field A of characteristic zero, the theory of 𝒟-fields has a model companion 𝒟-CF 0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.

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Existentially closed fields with<i>G</i>-derivations

2016-04-13

Daniel Max Hoffmann , Piotr Kowalski

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Derivations of the Frobenius map

2005-03-01

Piotr Kowalski

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Generalized Hasse-Schmidt varieties and their jet spaces

2011-02-03

Rahim Moosa , Thomas Scanlon

This work provides a unified formalism for studying difference and (Hasse-) differential algebraic geometry, by introducing a theory of "iterative Hasse rings and schemes". As an application, Hasse jet spaces … This work provides a unified formalism for studying difference and (Hasse-) differential algebraic geometry, by introducing a theory of "iterative Hasse rings and schemes". As an application, Hasse jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse variety is determined by its jet spaces at a point.

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Geometric axioms for existentially closed Hasse fields

2005-05-18

Piotr Kowalski

Model theory of fields with virtually free group actions

2018-07-26

Özlem Beyarslan , Piotr Kowalski

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The definable multiplicity property and generic automorphisms

2000-12-01

Hirotaka Kikyo , Anand Pillay

The Manin–Mumford conjecture and the model theory of difference fields

2001-09-01

Ehud Hrushovski

Tannakian Formalism Over Fields with Operators

2012-09-26

Moshe Kamensky

Journal Article Tannakian Formalism Over Fields with Operators Get access Moshe Kamensky Moshe Kamensky Department of Mathematics, University of Notre-Dame, Notre-Dame, IN, USA Correspondence to be sent to: [email protected] Search … Journal Article Tannakian Formalism Over Fields with Operators Get access Moshe Kamensky Moshe Kamensky Department of Mathematics, University of Notre-Dame, Notre-Dame, IN, USA Correspondence to be sent to: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2013, Issue 24, 2013, Pages 5571–5622, https://doi.org/10.1093/imrn/rns190 Published: 26 September 2012 Article history Received: 01 January 2012 Revision received: 09 July 2012 Accepted: 13 July 2012 Published: 26 September 2012

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THE THEORY OF COMMUTATIVE FORMAL GROUPS OVER FIELDS OF FINITE CHARACTERISTIC

1963-12-31

Yu. I. Manin

CONTENTS Introduction Chapter I. Formal groups and Dieudonné modules; basic concepts 1. Groups in categories 2. Algebraic and formal groups. Bialgebras 3. The structure of commutative artinian groups 4. The … CONTENTS Introduction Chapter I. Formal groups and Dieudonné modules; basic concepts 1. Groups in categories 2. Algebraic and formal groups. Bialgebras 3. The structure of commutative artinian groups 4. The Dieudonné module of a formal group 5. Comments Chapter II. Dieudonné modules; classification up to isogeny 1. Reduction of the problem 2. Modules over the ring A 3. A technical result 4. Classification of formal groups up to isogeny 5. Comments Chapter III. Dieudonné modules; classification up to isomorphism 1. Statement of the problem 2. Auxiliary results 3. The algebraic structure on the module space 4. The structure of isosimple modules; subsidiary reduction 5. The structure of isosimple modules; proof of the first finiteness theorem 6. The second finiteness theorem 7. Cyclic isosimple modules; the component of maximal dimension 8. Classification of two-dimensional modules 9. Comments Chapter IV. Algebroid formal groups and abelian varieties 1. General results 2. The formal structure of abelian varieties; preliminary reduction 3. The formal structure of abelian varieties; the fundamental theorem 4. Weakly algebroid groups 5. Remarks and examples 6. Comments References

On Schanuel's Conjectures

1971-03-01

James Ax

MODEL THEORY OF DIFFERENCE FIELDS, II: PERIODIC IDEALS AND THE TRICHOTOMY IN ALL CHARACTERISTICS

2002-08-02

Zoé Chatzidakis , Ehud Hrushovski , Ya’acov Peterzil

We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the … We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

The model theory of differential fields revisited

1976-09-01

Carol Wood

Formal Groups and Applications

2012-12-07

Michiel Hazewinkel

This book is a comprehensive treatment of the theory of formal groups and its numerous applications in several areas of mathematics. The seven chapters of the book present basics and … This book is a comprehensive treatment of the theory of formal groups and its numerous applications in several areas of mathematics. The seven chapters of the book present basics and main results of the theory, as well as very important applications in algebraic topology, number theory, and algebraic geometry. Each chapter ends with several pages of historical and bibliographic summary. One prerequisite for reading the book is an introductory graduate algebra course, including certain familiarity with category theory.

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Existentially closed fields with finite group actions

2017-11-24

Daniel Max Hoffmann , Piotr Kowalski

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Separably closed fields with higher derivations I

1995-09-01

Margit Messmer , Carol Wood

Abstract We define a complete theory SHF e of separably closed fields of finite invariant e (=degree of imperfection) which carry an infinite stack of Hasse-derivations. We show that SHF … Abstract We define a complete theory SHF e of separably closed fields of finite invariant e (=degree of imperfection) which carry an infinite stack of Hasse-derivations. We show that SHF e has quantifier elimination and eliminates imaginaries.

Some Topics in Differential Algebraic Geometry I: Analytic Subgroups of Algebraic Groups

1972-10-01

James Ax

Jet spaces of varieties over differential and difference fields

2003-12-01

Anand Pillay , Martin Ziegler

Model theoretic dynamics in Galois fashion

2019-02-19

Daniel Max Hoffmann

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Jets via Hasse-Schmidt Derivations

2004-07-07

Paul Vojta

This note is intended to provide a general reference for jet spaces and jet differentials, valid in maximal generality (at the level of EGA). The approach is rather concrete, using … This note is intended to provide a general reference for jet spaces and jet differentials, valid in maximal generality (at the level of EGA). The approach is rather concrete, using Hasse-Schmidt (divided) higher differentials. Discussion of projectivized jet spaces (as in Green and Griffiths (1980)) is included.

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Schanuel's conjecture for non-isoconstant elliptic curves over function fields

2008-05-22

Daniel Bertrand

We discuss functional and number theoretic extensions of Schanuel's conjecture, with special emphasis on the study of elliptic integrals of the third kind. We discuss functional and number theoretic extensions of Schanuel's conjecture, with special emphasis on the study of elliptic integrals of the third kind.

A note on a theorem of Ax

2008-07-08

Piotr Kowalski

Separably closed fields

2009-03-13

Françoise Delon

Integrating Hasse–Schmidt derivations

2014-05-16

Daniel Max Hoffmann , Piotr Kowalski

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The Mordell-Lang conjecture for function fields

1996-01-01

Ehud Hrushovski

We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in … We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Geometry on Arc Spaces of Algebraic Varieties

2001-01-01

Jan Denef , François Loeser

Geometric Stability Theory

1996-09-12

Anand Pillay

Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine … Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.

Notes on the stability of separably closed fields

1979-09-01

Carol Wood

Abstract The stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are … Abstract The stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in §3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some basic facts about model completeness [4], stability [7], separably closed fields [2] or [3], and (for §3 only) differential fields [8].

Geometric characterizations of existentially closed fields with operators

2004-10-01

David A. Pierce

This paper concerns the basic model-theory of fields of arbitrary characteristic with operators.Simplified geometric axioms are given for the model-companion of the theory of fields with a derivation.These axioms generalize … This paper concerns the basic model-theory of fields of arbitrary characteristic with operators.Simplified geometric axioms are given for the model-companion of the theory of fields with a derivation.These axioms generalize to the case of several commuting derivations.Let a D-field be a field with a derivation or a difference-operator, called D. The theory of D-fields is companionable.The existentially closed D-fields can be characterized geometrically without distinguishing the two cases in which D can fall.The class of existentially closed fields with a derivation and a difference-operator is elementary only in characteristic 0.

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Generic automorphisms of separably closed fields

2001-07-01

Zoé Chatzidakis

We show that the class of separably closed fields with a generic automorphism is an elementary class, whose theory is model complete in a natural extension of the language of … We show that the class of separably closed fields with a generic automorphism is an elementary class, whose theory is model complete in a natural extension of the language of fields with an automorphism. We describe the completions of this theory and obtain some results on types, imaginaries, and modularity.

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Profinite Groups

2000-01-01

Luis Ribes , Pavel Zalesskii

The Elementary Theory of the Frobenius Automorphisms

2004-06-25

Ehud Hrushovski

A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of … A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity. In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field. This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the latter. The setting for the proof is the geometry of difference varieties of transformal dimension zero; these generalize algebraic varieties, and are shown to have a rich structure, only partly explicated here. Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations.

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Zariski Geometries

2010-02-01

Boris Zilber

We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets. We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.

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The Arithmetic of Elliptic Curves

1986-01-01

Joseph H. Silverman

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Reducts of (C, +, ·) which contain +

1990-09-01

David R. Marker , Anthony L. Pillay

Abstract We show that the structure (C, +, ·) has no proper non locally modular reducts which contain +. In other words, if X ⊂ C n is constructible and … Abstract We show that the structure (C, +, ·) has no proper non locally modular reducts which contain +. In other words, if X ⊂ C n is constructible and not definable in the module structure (C, +, λ a ) a Є C (where λ a denotes multiplication by a ) then multiplication is definable in ( C , +, X ).

Differential Algebraic Groups of Finite Dimension

1992-01-01

Alexandru Buium

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Difference fields and descent in algebraic dynamics. I

2008-10-01

Zoé Chatzidakis , Ehud Hrushovski

Abstract We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In … Abstract We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line. The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics.

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Model theory of fields with free operators in positive characteristic

2019-06-05

Özlem Beyarslan , Daniel Max Hoffmann , Moshe Kamensky +1 more

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Invariant varieties for polynomial dynamical systems

2013-10-25

Alice Medvedev , Thomas Scanlon

We study algebraic dynamical systems (and, more generally, σ-varieties)given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials.More precisely, we find a nearly … We study algebraic dynamical systems (and, more generally, σ-varieties)given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials.More precisely, we find a nearly canonical way to write a polynomial as a composition of "clusters" from which one may easily read off possible compositional identities.Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphismis a polynomial of degree at least two that is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and X ⊆ A 2C is an irreducible curve that is invariant under the action of (x, y) → (f (x), f (y)) and projects dominantly in both directions, then X must be the graph of a polynomial that commutes with f under composition.As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius.We also show that in models of ACFA0, a disintegrated set defined by σ(x) = f (x) for a polynomial f has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skewconjugacy class of f is defined over a fixed field of a power of σ, and that nonorthogonality between two such sets is definable in families if the skewconjugacy class of f is defined over a fixed field of a power of σ.

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The theory of the exponential differential equations of semiabelian varieties

2009-07-16

Jonathan Kirby

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Arithmetic Analogues of Derivations

1997-12-01

Alexandru Buium

Schanuel property for additive power series

2011-10-06

Piotr Kowalski

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Model Theory and Algebraic Geometry

1998-01-01

Infinitesimal Group Scheme Actions on Finite Field Extensions

1976-01-01

Stephen U. Chase

Generic structures and simple theories

1998-11-01

Zoé Chatzidakis , Anthony L. Pillay

Functional transcendence via o-minimality

2015-08-05

Jonathan Pila

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Jacobi’s bound for the order of systems of first order differential equations

1970-01-01

Barbara A. Lando

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 1 comma ellipsis comma upper A Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 1 comma ellipsis comma upper A Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_1}, \ldots ,{A_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a system of differential polynomials in the differential indeterminates <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis 1 right-parenthesis Baseline comma ellipsis comma y Superscript left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{y^{(1)}}, \ldots ,{y^{(n)}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an irreducible component of the differential variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M left-parenthesis upper A 1 comma ellipsis comma upper A Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}({A_1}, \ldots ,{A_n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension script upper M equals 0"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\dim \mathcal {M} = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there arises the question of securing an upper bound for the order of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the orders <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r Subscript i j"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{r_{ij}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript i"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis j right-parenthesis"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>j</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{y^{(j)}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It has been conjectured that the Jacobi number <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J equals upper J left-parenthesis r Subscript i j Baseline right-parenthesis equals max left-brace sigma-summation Underscript i equals 1 Overscript n Endscripts r Subscript i j Sub Subscript i Subscript Baseline colon j 1 comma ellipsis comma j Subscript n Baseline is a permutation of 1 comma ellipsis comma n right-brace"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:munderover> <mml:mo movablelimits="false">∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:munderover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>j</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>j</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>j</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> is a permutation of 1,</mml:mtext> </mml:mrow> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">J = J({r_{ij}}) = \max \{\sum \limits _{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text { is a permutation of 1,}} \ldots ,n \}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> provides such a bound. In this paper <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained as a bound for systems consisting of first order polynomials. Differential kernels are employed in securing the bound, with the theory of kernels obtained in a manner analogous to that of difference kernels as given by R. M. Cohn.

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