Joint IAS/PU Number Theory
Polynomial Bounds for Birch's Theorem on Forms
Consider a collection of forms of odd degree with rational coefficients. Birch proved in 1957 that if the number of variables is sufficiently large, then the forms must have a nontrivial rational zero. The bounds resulting from Birch's proof, however, are so large that he has described them as "not even astronomical". We prove that, for any fixed odd degree, the number of variables may be taken polynomial in the number of equations. This was previously known only in degree three, by a result of Schmidt from 1982. We will review Birch's original argument, discuss a stronger result by Schmidt and sketch the proof of our theorem. Joint work with Andrew Snowden and Tamar Ziegler.
Date & Time
February 26, 2026 | 3:30pm – 4:30pm
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02/26/2026 15:30
02/26/2026 16:30
Joint IAS/PU Number Theory
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Topic: Polynomial Bounds for Birch's Theorem on Forms
Speakers: Amichai Lampert, University of Michigan
More: https://www.ias.edu/math/events/joint-iaspu-number-theory-8
Consider a collection of forms of odd degree with rational
coefficients. Birch proved in 1957 that if the number of variables is
sufficiently large, then the forms must have a nontrivial rational
zero. The bounds resulting from Birch's proof, however, are so large
that he has described them as "not even astronomical". We prove that,
for any fixed odd degree, the number of variables may be taken
polynomial in the number of equations. This was previously known only
in degree three, by a result of Schmidt from 1982. We will review
Birch's original argument, discuss a stronger result by Schmidt and
sketch the proof of our theorem. Joint work with Andrew Snowden and
Tamar Ziegler.
Simonyi 101 and Remote Access
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Location
Simonyi 101 and Remote AccessSpeakers
Amichai Lampert, University of Michigan