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Improve documentation of discrete valuations #39612

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Improve documentation of discrete valuations #39612

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c11e56d
Clarify runtime of augmentation_chain
saraedum Mar 1, 2025
ca46d4e
Explain how to reconstruct the sequence of key polynomials without _b…
saraedum Mar 1, 2025
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6 changes: 6 additions & 0 deletions src/sage/rings/valuation/augmented_valuation.py
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Original file line number Diff line number Diff line change
Expand Up @@ -433,6 +433,12 @@ def augmentation_chain(self):
r"""
Return a list with the chain of augmentations down to the underlying :mod:`Gauss valuation <sage.rings.valuation.gauss_valuation>`.

.. NOTE::

This method runs in time linear in the length of the chain (though
the printed representation might seem to indicate that it takes
quadratic time to construct the chain.)

EXAMPLES::

sage: R.<x> = QQ[]
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25 changes: 24 additions & 1 deletion src/sage/rings/valuation/developing_valuation.py
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Expand Up @@ -40,7 +40,7 @@
[x + 1, 1]
"""
# ****************************************************************************
# Copyright (C) 2013-2017 Julian Rüth <[email protected]>
# Copyright (C) 2013-2025 Julian Rüth <[email protected]>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
Expand Down Expand Up @@ -102,6 +102,29 @@ def phi(self):
sage: v = GaussValuation(S) # needs sage.libs.ntl
sage: v.phi() # needs sage.libs.ntl
(1 + O(2^5))*x

Use
:meth:`~sage.rings.valuation.inductive_valuation.InductiveValuation.augmentation_chain`
to obtain the sequence of key polynomials of an
:class:`~sage.rings.valuation.inductive_valuation.InductiveValuation`::

sage: R.<x> = QQ[]
sage: v = GaussValuation(R, QQ.valuation(2))
sage: v = v.augmentation(x, 1)
sage: v = v.augmentation(x^2 + 2*x + 4, 3)

sage: v
[ Gauss valuation induced by 2-adic valuation, v(x) = 1, v(x^2 + 2*x + 4) = 3 ]

sage: [w.phi() for w in v.augmentation_chain()[:-1]]
[x^2 + 2*x + 4, x]

A similar approach can be used to obtain the key polynomials and their
corresponding valuations::

sage: [(w.phi(), w.mu()) for w in v.augmentation_chain()[:-1]]
[(x^2 + 2*x + 4, 3), (x, 1)]

"""
return self._phi

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