This package implements most of Mac Lane's algorithms [1,2] and related structures to represent discrete valuations and discrete pseudo-valuations on rings in Sage.
This package is now obsolete. It has been merged completely into Sage, see https://trac.sagemath.org/ticket/21869.
The package should run on an unmodified Sage 8.0 and can be imported with
sage: from mac_lane import *
To run the included tests, execute sage -tp --optional=sage,standalone mac_lane/.
Valuations can be defined conveniently on some Sage rings such as p-adic rings and function fields.
Valuations on number fields can be easily specified if they uniquely extend the valuation of a rational prime:
sage: v = pAdicValuation(QQ, 2)
sage: v(1024)
10
They are normalized such that the rational prime has valuation 1.
sage: K.<a> = NumberField(x^2 + x + 1)
sage: v = pAdicValuation(K, 2)
sage: v(1024)
10
If there are multiple valuations over a prime, they can be obtained by extending a valuation from a smaller ring.
sage: K.<a> = NumberField(x^2 + x + 1)
sage: v = pAdicValuation(QQ, 7)
sage: v.extensions(K)
[[ 7-adic valuation, v(x + 3) = 1 ]-adic valuation,
[ 7-adic valuation, v(x + 5) = 1 ]-adic valuation]
sage: w,ww = _
sage: w(a + 3), ww(a + 3)
(1, 0)
sage: w(a + 5), ww(a + 5)
(0, 1)
Similarly, valuations can be defined on function fields:
sage: K.<x> = FunctionField(QQ)
sage: v = FunctionFieldValuation(K, x)
sage: v(1/x)
-1
sage: v = FunctionFieldValuation(K, 1/x)
sage: v(1/x)
1
On extensions of function fields, valuations can be specified explicitly by providing a prime on the underlying rational function field when the extension is unique::
sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: v = FunctionFieldValuation(L, x)
sage: v(x)
1
Valuations can also be extended from smaller function fields.
sage: K.<x> = FunctionField(QQ)
sage: v = FunctionFieldValuation(K, x - 4)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: v.extensions(L)
[ (x - 4)-adic valuation, v(y - 2) = 1 ]-adic valuation,
[ (x - 4)-adic valuation, v(y + 2) = 1 ]-adic valuation]
Internally, all the above is backed by the algorithms described in [1,2]. Let
us consider the extensions of FunctionFieldValuation(K, x - 4) to the field
L above to outline how this works internally.
First, the valuation on K is induced by a valuation on ℚ[x]. To construct this valuation, we start from the trivial valuation on ℚ and consider its induced Gauss valuation on ℚ[x], i.e., the valuation that assigns to a polynomial the minimum of the coefficient valuations.
sage: R.<x> = QQ[]
sage: v = TrivialValuation(QQ)
sage: v = GaussValuation(R, v)
The Gauss valuation can be augmented by specifying that x - 4 has valuation 1.
sage: v = v.augmentation(x - 4, 1); v
[ Gauss valuation induced by Trivial valuation on Rational Field, v(x - 4) = 1 ]
This valuation then extends uniquely to the fraction field.
sage: K.<x> = FunctionField(QQ)
sage: v = v.extension(K); v
(x - 4)-adic valuation
Over the function field we repeat the above process, i.e., we define the Gauss valuation induced by it and augment it to approximate an extension to L.
sage: R.<y> = K[]
sage: w = GaussValuation(R, v)
sage: w = w.augmentation(y - 2, 1); w
[ Gauss valuation induced by (x - 4)-adic valuation, v(y - 2) = 1 ]
sage: L.<y> = K.extension(y^2 - x)
sage: ww = w.extension(L); ww
[ (x - 4)-adic valuation, v(y - 2) = 1 ]-adic valuation
In the previous example the final valuation ww is not merely given by evaluating w on the ring K[y].
sage: ww(y^2 - x)
+Infinity
sage: y = R.gen()
sage: w(y^2 - x)
1
Instead ww is given by a limit, i.e., an infinite sequence of augmentations of valuations.
sage: ww._base_valuation
[ Gauss valuation induced by (x - 4)-adic valuation, v(y - 2) = 1 , … ]
The terms of this infinite sequence are computed on demand.
sage: ww._base_valuation._approximation
[ Gauss valuation induced by (x - 4)-adic valuation, v(y - 2) = 1 ]
sage: ww(y - 1/4*x - 1)
2
sage: ww._base_valuation._approximation
[ Gauss valuation induced by (x - 4)-adic valuation, v(y - 1/4*x - 1) = 2 ]
Using the low-level interface we are not limited to classical valuations on function fields that correspond to points on the corresponding curves. Instead we can start with a non-trivial valuation on the field of constants.
sage: v = pAdicValuation(QQ, 2)
sage: R.<x> = QQ[]
sage: w = GaussValuation(R, v) # v is not trivial
sage: K.<x> = FunctionField(QQ)
sage: w = w.extension(K)
sage: w.residue_field()
Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 2 (using NTL)
The main tool underlying this package is an algorithm by Mac Lane to compute, starting from a Gauss valuation on a polynomial ring and a monic squarefree polynomial G, approximations to the limit valuation which send G to infinity.
sage: v = pAdicValuation(QQ, 2)
sage: R.<x> = QQ[]
sage: f = x^5 + 3*x^4 + 5*x^3 + 8*x^2 + 6*x + 12
sage: v.mac_lane_approximants(f)
[[ Gauss valuation induced by 2-adic valuation, v(x^2 + x + 1) = 3 ],
[ Gauss valuation induced by 2-adic valuation, v(x) = 1/2 ],
[ Gauss valuation induced by 2-adic valuation, v(x) = 1 ]]
Note that from these approximants one can already see the residual degrees and ramification indices of the corresponding extensions. The approximants can be pushed to arbitrary precision.
sage: sage: v.mac_lane_approximants(f, required_precision=10)
[[ Gauss valuation induced by 2-adic valuation, v(x^2 + 193*x + 13/21) = 10 ],
[ Gauss valuation induced by 2-adic valuation, v(x + 86) = 10 ],
[ Gauss valuation induced by 2-adic valuation, v(x) = 1/2, v(x^2 + 36/11*x + 2/17) = 11 ]]
Note that in the limit they are factors of f.
[1] Mac Lane, S. (1936). A construction for prime ideals as absolute values of an algebraic field. Duke Mathematical Journal, 2(3), 492-510.
[2] MacLane, S. (1936). A construction for absolute values in polynomial rings. Transactions of the American Mathematical Society, 40(3), 363-395.
[3] Rüth, J. (2014). Models of Curves and Valuations (PhD thesis) Chapter 4 "Mac Lane Valuations".