the space of real places of ℝ (x,y) ron brown jon merzel
DESCRIPTION
Our results: The space is actually path connected. For each (isomorphism class of) value group, the set of all corresponding places is dense. Some large collections of mutually homeomorphic subspaces are identified.TRANSCRIPT
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The Space of Real Places of ℝ(x,y)Ron BrownJon Merzel
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The Space M(ℝ(x,y)) Weakest topology making evaluation maps
continuous Subbasic “Harrison” sets of the form {: (f)∊(0, ∞)} where f ∊ ℝ(x,y) Well-known:
Compact Hausdorff Connected
Contains torus??? Disk???
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Our results:
The space is actually path connected. For each (isomorphism class of) value
group, the set of all corresponding places is dense.
Some large collections of mutually homeomorphic subspaces are identified.
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Method For be the place corresponding to the
composition of the place y↦0 from (x,y) to (x) with the place x ↦ c from (x) to .
The places form a “circle” in M((x,y)). M((x,y)) is a union of homeomorphic
“fibers” = M((x,y)):(x)=c}, one through each point of the circle.
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M(ℝ . Brown (1972) analyzes all extensions of
a complete discrete rank one valuation to a simple transcendental extension.
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How to represent M The elements of M are in bijection with
certain sequences where s either a real number, , or of the form
or for some rational number is a real number is a positive integer, or
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How to build a legitimate sequence
RepeatChoose ;Choose ℚ ;
Until (What if the loop is infinite? Coming!)
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Examplen0 2/3 17 3 21 13/6 -2π 2 13/32 03 + The sequence corresponds to a place with value group + ( = + )
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Infinite length sequences
If for all large then we set the length Otherwise . In both cases,
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How to picture M The set of possible pairs can be
pictured as follows: Break the real line at a rational and join
the two rays with a circle.
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Do this at every rational, and put in points at to get the bubble line.
Now, how to picture a sequence of ()’s. Each finite sequence with rational ’s has
a minimum possible , namely . Make that the first point of a new bubble
line.
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The sequence The infinite bedspring
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The points of M Finite sequences corresponding to points of
M can be pictured (uniquely) as points on the infinite bedspring.
Infinite sequences corresponding to points of M can be visualized (uniquely) as infinite “paths” through the infinite bedspring.
The topology on the bedspring (induced by the Harrision topology via the bijection) is a little technical.
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Path connectedness If you keep only the top or bottom half
of each circle, the “half-bubble line” becomes a linearly ordered set, and the induced topology is the order topology.
With that topology, the half-bubble line is homeomorphic to a closed interval on the real line.
Stitching together pieces corresponding to closed intervals gives us paths.
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Density The topology on the space of sequences
has the following property: Given any nonempty open set, there is a
finite sequence S=with rational ’s such that every admissible sequence beginning with S is in the open set.
Freedom choosing q’s & q’s generate Γ. Make sure valuation restricts properly.
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Self-similarity Look at all sequences with common
start with rational ’s and with where is fixed.
These all look the same!(Checking the topology is a messy case-by-case computation)
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Under the bijection between the set of legitmate signatures and M, this set of signatures corresponds to a certain subbasic open set, determined by a choice of an irreducible polynomial and a sufficiently large rational number.
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The bijection can be established via “strict systems” Closely related to the “saturated
distinguished chains” of Popescu, Khanduja et al