# Sage 3.4.2 Release Tour

Sage 3.4.2 was released on May 05, 2009. For the official, comprehensive release note, please refer to sage-3.4.2.txt. A nicely formatted version of this release tour can be found here. The following points are some of the foci of this release:

• Improve doctest coverage of the Sage library in anticipation of version 4.0.
• New features for symbolic logic.

## Algebra

• Comparison of ring coercion morphisms (Alex Ghitza) -- New comparison method __cmp__() for the class RingHomomorphism_coercion in sage/rings/morphism.pyx. The comparison method __cmp__(self, other) compares a ring coercion morphism self to other. Ring coercion morphisms never compare equal to any other data type. If other is a ring coercion morphism, the parents of self and other are compared. Here are some examples on comparing ring coercion morphisms:

```sage: f = ZZ.hom(QQ)
sage: g = ZZ.hom(ZZ)
sage: f == g
False
sage: f > g
True
sage: f < g
False
sage: h = Zmod(6).lift()
sage: f == h
False
sage: f = ZZ.hom(QQ)
sage: g = loads(dumps(f))
sage: f == g
True```
• Coercing factors into a common universe (Alex Ghitza) -- New method base_change(self, U) in the module sage/structure/factorization.py to allow the factorization self with its factors (including the unit part) coerced into the universe U. Here's an example for working with the new method base_change():

```sage: F = factor(2006)
sage: F.universe()
Integer Ring
sage: P.<x> = ZZ["x"]
sage: F.base_change(P).universe()
Univariate Polynomial Ring in x over Integer Ring```

## Basic Arithmetic

• Enhancements to symbolic logic (Chris Gorecki) -- This adds a number of utilities for working with symbolic logic:
1. sage/logic/booleval.py -- For evaluating boolean formulas.

2. sage/logic/boolformula.py -- For boolean evaluation of boolean formulas.

3. sage/logic/logicparser.py -- For creating and modifying parse trees of well-formed boolean formulas.

4. sage/logic/logictable.py -- For creating and printing truth tables associated with logical statements.

5. sage/logic/propcalc.py -- For propositional calculus.

Here are some examples for working with the new symbolic logic modules:
```sage: import sage.logic.propcalc as propcalc
sage: f = propcalc.formula("a&((b|c)^a->c)<->b")
sage: g = propcalc.formula("boolean<->algebra")
sage: (f&~g).ifthen(f)
((a&((b|c)^a->c)<->b)&(~(boolean<->algebra)))->(a&((b|c)^a->c)<->b)
sage: f.truthtable()

a      b      c      value
False  False  False  True
False  False  True   True
False  True   False  False
False  True   True   False
True   False  False  True
True   False  True   False
True   True   False  True
True   True   True   True```
• New function squarefree_divisors() (Robert Miller) -- The new function squarefree_divisors(x) in the module sage/rings/arith.py allows for iterating over the squarefree divisors (up to units) of the element x. Here, we assume that x is an element of any ring for which the function prime_divisors() works. Below are some examples for working with the new function squarefree_divisors():

```sage: list(squarefree_divisors(7))
[1, 7]
sage: list(squarefree_divisors(6))
[1, 2, 3, 6]
sage: list(squarefree_divisors(81))
[1, 3]```

## Combinatorics

• Make cartan_type a method rather than an attribute (Dan Bump) -- For the module sage/combinat/root_system/weyl_characters.py, cartan_type is now a method, not an attribute. For example, one can now invoke cartan_type as a method like so:

```sage: A2 = WeylCharacterRing("A2")
sage: A2([1,0,0]).cartan_type()
['A', 2]```

## Commutative Algebra

• Improved performance in MPolynomialRing_libsingular (Simon King) -- This provides some optimization of the method MPolynomialRing_libsingular.__call__(). In some cases, the efficiency is up to 19%. The following timing statistics are obtained using the machine sage.math:

```# BEFORE

sage: R = PolynomialRing(QQ,5,"x")
sage: S = PolynomialRing(QQ,6,"x")
sage: T = PolynomialRing(QQ,5,"y")
sage: U = PolynomialRing(GF(2),5,"x")
sage: p = R("x0*x1+2*x4+x3*x1^2")^4
sage: timeit("q = S(p)")
625 loops, best of 3: 321 µs per loop
sage: timeit("q = T(p)")
625 loops, best of 3: 348 µs per loop
sage: timeit("q = U(p)")
625 loops, best of 3: 435 µs per loop

# AFTER

sage: R = PolynomialRing(QQ,5,"x")
sage: S = PolynomialRing(QQ,6,"x")
sage: T = PolynomialRing(QQ,5,"y")
sage: U = PolynomialRing(GF(2),5,"x")
sage: p = R("x0*x1+2*x4+x3*x1^2")^4
sage: timeit("q = S(p)")
625 loops, best of 3: 316 µs per loop
sage: timeit("q = T(p)")
625 loops, best of 3: 281 µs per loop
sage: timeit("q = U(p)")
625 loops, best of 3: 392 µs per loop```

## Graph Theory

• Default edge color is black (Robert Miller) -- If only one edge of a graph is colored red, for example, then the remaining edges should be colored with black by default. Here's an example:
```sage: G = graphs.CompleteGraph(5)
sage: G.show(edge_colors={'red':[(0,1)]})``` ## Interfaces

• Split off the FriCAS interface from the Axiom interface (Mike Hansen, Bill Page) -- The FriCAS interface is now split off from the Axiom interface and can now be found in the module sage/interfaces/fricas.py.

## Modular Forms

• Vast speedup in P1List construction (John Cremona) -- This provides huge improvement in the P1List() constructor for Manin symbols. The efficiency gain can range from 27% up to 6x. Here are some timing statistics obtained using the machine sage.math:

```# BEFORE

sage: time P1List(100000)
CPU times: user 4.11 s, sys: 0.08 s, total: 4.19 s
Wall time: 4.19 s
The projective line over the integers modulo 100000
sage: time P1List(1000000)
CPU times: user 192.22 s, sys: 0.60 s, total: 192.82 s
Wall time: 192.84 s
The projective line over the integers modulo 1000000
sage: time P1List(1009*1013)
CPU times: user 31.20 s, sys: 0.05 s, total: 31.25 s
Wall time: 31.25 s
The projective line over the integers modulo 1022117
sage: time P1List(1000003)
CPU times: user 35.92 s, sys: 0.05 s, total: 35.97 s
Wall time: 35.97 s
The projective line over the integers modulo 1000003

# AFTER

sage: time P1List(100000)
CPU times: user 0.78 s, sys: 0.02 s, total: 0.80 s
Wall time: 0.80 s
The projective line over the integers modulo 100000
sage: time P1List(1000000)
CPU times: user 27.82 s, sys: 0.21 s, total: 28.03 s
Wall time: 28.02 s
The projective line over the integers modulo 1000000
sage: time P1List(1009*1013)
CPU times: user 21.59 s, sys: 0.04 s, total: 21.63 s
Wall time: 21.63 s
The projective line over the integers modulo 1022117
sage: time P1List(1000003)
CPU times: user 26.19 s, sys: 0.05 s, total: 26.24 s
Wall time: 26.24 s
The projective line over the integers modulo 1000003```

## Notebook

• Reduce the number of actions that trigger taking a snapshot (William Stein, Rob Beezer) -- Snapshots now need to be explicitly requested by clicking the save button. This greatly reduces many unnecessary snapshots.

## Number Theory

• Enhanced function prime_pi() for counting primes (R. Andrew Ohana) -- The improved function prime_pi() in sage/functions/prime_pi.pyx implements the prime counting function pi(n). Essentially, prime_pi(n) counts the number of primes less than or equal to n. Here are some examples:

```sage: prime_pi(10)
4
sage: prime_pi(100)
25
sage: prime_pi(-10)
0
sage: prime_pi(-0.5)
0
sage: prime_pi(10^10)
455052511```
• Action of the Galois group on cusps (William Stein) -- New method galois_action() in sage/modular/cusps.py for computing action of the Galois group on cusps for congruence subgroups. The relevant algorithm here is taken from section 1.3 of the following text:

• S. Glenn. Arithmetic on Modular Curves. Progress in Mathematics, volume 20, Birkhauser, 1982.

Here are some examples for working with galois_action():

```sage: Cusp(1/10).galois_action(3, 50)
1/170
sage: Cusp(oo).galois_action(3, 50)
Infinity
sage: Cusp(0).galois_action(3, 50)
0```
• Finding elliptic curves with prescribed reduction over QQ (John Cremona) -- New function EllipticCurves_with_good_reduction_outside_S() for constructing elliptic curves with good reduction outside a finite set of primes. This essentially implements the algorithm presented in the paper, but currently only over QQ:

• J. Cremona and M. Lingham. Finding all elliptic curves with good reduction outside a given set of primes. Experimental Mathematics, 16(3):303--312, 2007.
Here are some examples for working with this new function:
```sage: EllipticCurves_with_good_reduction_outside_S([])
[]
sage: elist = EllipticCurves_with_good_reduction_outside_S()
sage: elist

[Elliptic Curve defined by y^2 = x^3 + 4*x over Rational Field,
Elliptic Curve defined by y^2 = x^3 - x over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 11*x - 14 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 11*x + 14 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 44*x - 112 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 44*x + 112 over Rational Field,
Elliptic Curve defined by y^2 = x^3 + x over Rational Field,
Elliptic Curve defined by y^2 = x^3 + x^2 + x + 1 over Rational Field,
Elliptic Curve defined by y^2 = x^3 + x^2 - 9*x + 7 over Rational Field,
Elliptic Curve defined by y^2 = x^3 + x^2 + 3*x - 5 over Rational Field,
Elliptic Curve defined by y^2 = x^3 + x^2 - 2*x - 2 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - x^2 + x - 1 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - x^2 - 9*x - 7 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - x^2 + 3*x + 5 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - x^2 - 2*x + 2 over Rational Field,
Elliptic Curve defined by y^2 = x^3 + x^2 - 3*x + 1 over Rational Field,
Elliptic Curve defined by y^2 = x^3 + x^2 - 13*x - 21 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 2*x over Rational Field,
Elliptic Curve defined by y^2 = x^3 + 8*x over Rational Field,
Elliptic Curve defined by y^2 = x^3 + 2*x over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 8*x over Rational Field,
Elliptic Curve defined by y^2 = x^3 - x^2 - 3*x - 1 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - x^2 - 13*x + 21 over Rational Field]
sage: len(elist)
24
sage: ', '.join([e.label() for e in elist])
'32a1, 32a2, 32a3, 32a4, 64a1, 64a2, 64a3, 64a4, 128a1, 128a2, 128b1, 128b2, 128c1, 128c2, 128d1, 128d2, 256a1, 256a2, 256b1, 256b2, 256c1, 256c2, 256d1, 256d2'```
• Make elliptic curves over the mod rings Z/pZ behave like elliptic curves over the finite fields GF(p) (Alex Ghitza) -- Elliptic curves over Z/NZ for prime N are now treated as being over a finite field. For example,

```sage: F = Zmod(101)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field'>```

However, if N is composite, then elliptic curves over Z/NZ are treated as being of the type "generic elliptic curve". For example,

```sage: F = Zmod(95)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic'>```
• Clean up sage/schemes/elliptic_curves/ell_generic.py (Alex Ghitza) -- A lot of code in the module sage/schemes/elliptic_curves/ell_generic.py has been moved around and cleaned up. In particular, all methods relating to twists from ell_generic.py have been moved to sage/schemes/elliptic_curves/ell_field.py, including the alias base_field = base_ring. We now have change_ring being an alias for base_extend, since the two have the exact same functionality and equivalent code. And the standalone function Hasse_bounds has been moved from ell_generic.py to sage/schemes/plane_curves/projective_curve.py.

## Packages

• Upgrade Cython to version 0.11.1 latest upstream release (Robert Bradshaw) -- Based on Pyrex, Cython is a language that closely resembles Python and developed for writing C extensions for Python. For critical functionalities and performance, Sage uses Cython to generate very efficient C code from Cython code, for wrapping external C libraries, and for fast C modules that speed up the execution of Python code.

• Upgrade MPIR to version 1.1.1 latest upstream release (Michael Abshoff) -- MPIR is a library for multiprecision integers and rationals based on the GMP project. Among other things, MPIR aims to provide native build capability under Windows.

• Move DSage to its own spkg (William Stein) -- The Distributed Sage framework (DSage) contained in sage/dsage is now packaged as a self-contained spkg. DSage allows for distributed computing from within Sage.

• Update the FLINT spkg (Michael Abshoff) -- The new FLINT spkg is flint-1.2.4.p2.spkg and fixes spkg-check on OS X 64-bit.

• Update the Maxima spkg (Michael Abshoff) -- The Lisp implementation Clisp needs a stack size larger than many systems provide, i.e. 8KB. When Clisp is used as the Lisp implementation for Maxima, then Maxima can randomly fail to build if the stack size is not raised. The updated Maxima spkg maxima-5.16.3.p2.spkg sets the stack size to 32KB for Clisp.

ReleaseTours/sage-3.4.2 (last edited 2009-12-27 10:09:30 by Minh Nguyen)