Undecidability in function fields of positive characteristic

Kirsten Eisenträger; Alexandra Shlapentokh

doi:10.1093/imrn/rnp079

Undecidability in function fields of positive characteristic

Kirsten Eisenträger
, Alexandra Shlapentokh

Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

We prove that the first-order theory of any function field K of characteristic p > 2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. The proof uses a result by Moret-Bailly about ranks of elliptic curves over function fields.

Original language	English (US)
Pages (from-to)	4051-4086
Number of pages	36
Journal	International Mathematics Research Notices
Volume	2009
Issue number	21
DOIs	https://doi.org/10.1093/imrn/rnp079
State	Published - 2009

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1093/imrn/rnp079

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title = "Undecidability in function fields of positive characteristic",

abstract = "We prove that the first-order theory of any function field K of characteristic p > 2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. The proof uses a result by Moret-Bailly about ranks of elliptic curves over function fields.",

author = "Kirsten Eisentr{\"a}ger and Alexandra Shlapentokh",

note = "Funding Information: The authors would like to thank Arno Fehm and Stephen Simpson for helpful discussions leading to Corollary 5.1. The authors would also like to thank the referee for numerous corrections and helpful comments. K. Eisentr{\"a}ger was partially supported by NSF grant DMS-0801123 and a grant from the John Templeton Foundation. A. Shlapentokh was partially supported by NSF grants DMS-0354907 and DMS-0650927 as well as by a grant from the John Templeton Foundation.",

year = "2009",

doi = "10.1093/imrn/rnp079",

language = "English (US)",

volume = "2009",

pages = "4051--4086",

journal = "International Mathematics Research Notices",

issn = "1073-7928",

publisher = "Oxford University Press",

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TY - JOUR

T1 - Undecidability in function fields of positive characteristic

AU - Eisenträger, Kirsten

AU - Shlapentokh, Alexandra

N1 - Funding Information: The authors would like to thank Arno Fehm and Stephen Simpson for helpful discussions leading to Corollary 5.1. The authors would also like to thank the referee for numerous corrections and helpful comments. K. Eisenträger was partially supported by NSF grant DMS-0801123 and a grant from the John Templeton Foundation. A. Shlapentokh was partially supported by NSF grants DMS-0354907 and DMS-0650927 as well as by a grant from the John Templeton Foundation.

PY - 2009

Y1 - 2009

N2 - We prove that the first-order theory of any function field K of characteristic p > 2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. The proof uses a result by Moret-Bailly about ranks of elliptic curves over function fields.

AB - We prove that the first-order theory of any function field K of characteristic p > 2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. The proof uses a result by Moret-Bailly about ranks of elliptic curves over function fields.

UR - https://www.scopus.com/pages/publications/84864115282

UR - https://www.scopus.com/pages/publications/84864115282#tab=citedBy

U2 - 10.1093/imrn/rnp079

DO - 10.1093/imrn/rnp079

M3 - Article

AN - SCOPUS:84864115282

SN - 1073-7928

VL - 2009

SP - 4051

EP - 4086

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 21

ER -

Undecidability in function fields of positive characteristic

Abstract

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