Coverage for local/lib/python2.7/site-packages/sage/interfaces/macaulay2.py : 6%

r""" 

Interface to Macaulay2 

.. NOTE:: 

You must have ``Macaulay2`` installed on your computer 

for this interface to work. Macaulay2 is not included with Sage, 

but you can obtain it from <http://www.math.uiuc.edu/Macaulay2/>. 

Note additional optional Sage packages are required. 

Sage provides an interface to the Macaulay2 computational algebra 

system. This system provides extensive functionality for commutative 

algebra. You do not have to install any optional packages. 

The Macaulay2 interface offers three pieces of functionality: 

- ``Macaulay2_console()`` -- A function that dumps you 

into an interactive command-line Macaulay2 session. 

- ``Macaulay2(expr)`` -- Evaluation of arbitrary Macaulay2 

expressions, with the result returned as a string. 

- ``Macaulay2.new(expr)`` -- Creation of a Sage object that wraps a 

Macaulay2 object. This provides a Pythonic interface to Macaulay2. For 

example, if ``f=Macaulay2.new(10)``, then ``f.gcd(25)`` returns the 

GCD of `10` and `25` computed using Macaulay2. 

EXAMPLES:: 

sage: print(macaulay2('3/5 + 7/11')) # optional - macaulay2 

68 

-- 

55 

sage: f = macaulay2('f = i -> i^3') # optional - macaulay2 

sage: f # optional - macaulay2 

f 

sage: f(5) # optional - macaulay2 

125 

sage: R = macaulay2('ZZ/5[x,y,z]') # optional - macaulay2 

sage: print(R) # optional - macaulay2 

ZZ 

--[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1] 

5 {GRevLex => {3:1} } 

{Position => Up } 

sage: x = macaulay2('x') # optional - macaulay2 

sage: y = macaulay2('y') # optional - macaulay2 

sage: print((x+y)^5) # optional - macaulay2 

5 5 

x + y 

sage: parent((x+y)^5) # optional - macaulay2 

Macaulay2 

sage: R = macaulay2('QQ[x,y,z,w]') # optional - macaulay2 

sage: f = macaulay2('x^4 + 2*x*y^3 + x*y^2*w + x*y*z*w + x*y*w^2 + 2*x*z*w^2 + y^4 + y^3*w + 2*y^2*z*w + z^4 + w^4') # optional - macaulay2 

sage: print(f) # optional - macaulay2 

4 3 4 4 2 3 2 2 2 4 

x + 2x*y + y + z + x*y w + y w + x*y*z*w + 2y z*w + x*y*w + 2x*z*w + w 

sage: g = f * macaulay2('x+y^5') # optional - macaulay2 

sage: print(g.factor()) # optional - macaulay2 

4 3 4 4 2 3 2 2 2 4 5 

(x + 2x*y + y + z + x*y w + y w + x*y*z*w + 2y z*w + x*y*w + 2x*z*w + w )(y + x) 

AUTHORS: 

- Kiran Kedlaya and David Roe (2006-02-05, during Sage coding sprint) 

- William Stein (2006-02-09): inclusion in Sage; prompt uses regexp, 

calling of Macaulay2 functions via __call__. 

- William Stein (2006-02-09): fixed bug in reading from file and 

improved output cleaning. 

- Kiran Kedlaya (2006-02-12): added ring and ideal constructors, 

list delimiters, is_Macaulay2Element, sage_polystring, 

__floordiv__, __mod__, __iter__, __len__; stripped extra 

leading space and trailing newline from output. 

.. TODO:: 

Get rid of all numbers in output, e.g., in ideal function below. 

""" 

#***************************************************************************** 

# Copyright (C) 2006 Kiran S. Kedlaya <kedlaya@mit.edu> 

# David Roe <roed@mit.edu> 

# William Stein <wstein@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function 

import os 

from sage.interfaces.expect import (Expect, ExpectElement, ExpectFunction, 

AsciiArtString) 

from sage.misc.multireplace import multiple_replace 

from sage.interfaces.tab_completion import ExtraTabCompletion 

from sage.docs.instancedoc import instancedoc 

import re 

def remove_output_labels(s): 

r""" 

Remove output labels of Macaulay2 from a string. 

- s: output of Macaulay2 

- s: string 

Returns: the input string with `n` symbols removed from the beginning of 

each line, where `n` is the minimal number of spaces or symbols of 

Macaulay2 output labels (looking like 'o39 = ') present on every non-empty 

line. 

Return type: string 

.. note:: 

If ``s`` consists of several outputs and their lables have 

different width, it is possible that some strings will have leading 

spaces (or maybe even pieces of output labels). However, this 

function will try not cut any messages. 

EXAMPLES:: 

sage: from sage.interfaces.macaulay2 import remove_output_labels 

sage: output = 'o1 = QQ [x, y]\n\no1 : PolynomialRing\n' 

sage: remove_output_labels(output) 

'QQ [x, y]\n\nPolynomialRing\n' 

""" 

label = re.compile("^o[0-9]+ (=|:) |^ *") 

lines = s.split("\n") 

matches = [label.match(l) for l in lines if l != ""] 

if len(matches) == 0: 

return s 

else: 

n = min(m.end() - m.start() for m in matches) 

return "\n".join(l[n:] for l in lines) 

PROMPT = "_EGAS_ : " 

class Macaulay2(ExtraTabCompletion, Expect): 

""" 

Interface to the Macaulay2 interpreter. 

""" 

def __init__(self, maxread=None, script_subdirectory=None, 

logfile=None, server=None,server_tmpdir=None): 

""" 

Initialize a Macaulay2 interface instance. 

We replace the standard input prompt with a strange one, so that 

we do not catch input prompts inside the documentation. 

We replace the standard input continuation prompt, which is 

just a bunch of spaces and cannot be automatically detected in a 

reliable way. This is necessary for allowing commands that occupy 

several strings. 

We also change the starting line number to make all the output 

labels to be of the same length. This allows correct stripping of 

the output of several commands. 

TESTS:: 

sage: macaulay2 == loads(dumps(macaulay2)) 

True 

""" 

init_str = ( 

# Prompt changing commands 

"""ZZ#{Standard,Core#"private dictionary"#"InputPrompt"} = lineno -> "%s";""" % PROMPT + 

"""ZZ#{Standard,Core#"private dictionary"#"InputContinuationPrompt"} = lineno -> "%s";""" % PROMPT + 

# Also prevent line wrapping in Macaulay2 

"printWidth = 0;" + 

# And make all output labels to be of the same width 

"lineNumber = 10^9;") 

Expect.__init__(self, 

name = 'macaulay2', 

prompt = PROMPT, 

command = "M2 --no-debug --no-readline --silent -e '%s'" % init_str, 

server = server, 

server_tmpdir = server_tmpdir, 

script_subdirectory = script_subdirectory, 

verbose_start = False, 

logfile = logfile, 

eval_using_file_cutoff=500) 

# Macaulay2 provides no "clear" function. However, Macaulay2 does provide 

# garbage collection; since expect automatically reuses variable names, 

# garbage collection in Sage properly sets up garbage collection in 

# Macaulay2. 

def __reduce__(self): 

""" 

Used in serializing an Macaulay2 interface. 

EXAMPLES:: 

sage: rlm2, t = macaulay2.__reduce__() 

sage: rlm2(*t) 

Macaulay2 

""" 

return reduce_load_macaulay2, tuple([]) 

def _read_in_file_command(self, filename): 

""" 

Load and *execute* the content of ``filename`` in Macaulay2. 

INPUT: 

- filename: the name of the file to be loaded and executed 

(type: string) 

OUTPUT: 

Returns Macaulay2 command loading and executing commands in 

``filename``, that is, ``'load "filename"'``. 

Return type: string 

TESTS:: 

sage: filename = tmp_filename() 

sage: f = open(filename, "w") 

sage: _ = f.write("sage_test = 7;") 

sage: f.close() 

sage: command = macaulay2._read_in_file_command(filename) 

sage: macaulay2.eval(command) # optional - macaulay2 

sage: macaulay2.eval("sage_test") # optional - macaulay2 

7 

sage: import os 

sage: os.unlink(filename) 

sage: macaulay2._read_in_file_command("test") 

'load "test"' 

sage: macaulay2(10^10000) == 10^10000 # optional - macaulay2 

True 

""" 

return 'load "%s"' % filename 

def __getattr__(self, attrname): 

""" 

EXAMPLES:: 

sage: gb = macaulay2.gb # optional - macaulay2 

sage: type(gb) # optional - macaulay2 

<class 'sage.interfaces.macaulay2.Macaulay2Function'> 

sage: gb._name # optional - macaulay2 

'gb' 

""" 

if attrname[:1] == "_": 

raise AttributeError 

return Macaulay2Function(self, attrname) 

def eval(self, code, strip=True, **kwds): 

""" 

Send the code x to the Macaulay2 interpreter and return the output 

as a string suitable for input back into Macaulay2, if possible. 

INPUT: 

- code -- str 

- strip -- ignored 

EXAMPLES:: 

sage: macaulay2.eval("2+2") # optional - macaulay2 

4 

""" 

code = code.strip() 

# TODO: in some cases change toExternalString to toString?? 

ans = Expect.eval(self, code, strip=strip, **kwds).strip('\n') 

if strip: 

ans = remove_output_labels(ans) 

return AsciiArtString(ans) 

def restart(self): 

r""" 

Restart Macaulay2 interpreter. 

TESTS:: 

sage: macaulay2.restart() # optional - macaulay2 

""" 

# If we allow restart to be called as a function, there will be 

# parasitic output 

self.eval("restart") 

def get(self, var): 

""" 

Get the value of the variable var. 

EXAMPLES:: 

sage: macaulay2.set("a", "2") # optional - macaulay2 

sage: macaulay2.get("a") # optional - macaulay2 

2 

""" 

return self.eval("describe %s"%var, strip=True) 

def set(self, var, value): 

""" 

Set the variable var to the given value. 

EXAMPLES:: 

sage: macaulay2.set("a", "2") # optional - macaulay2 

sage: macaulay2.get("a") # optional - macaulay2 

2 

""" 

cmd = '%s=%s;'%(var,value) 

ans = Expect.eval(self, cmd) 

if ans.find("stdio:") != -1: 

raise RuntimeError("Error evaluating Macaulay2 code.\nIN:%s\nOUT:%s"%(cmd, ans)) 

def _object_class(self): 

""" 

Returns the class of Macaulay2 elements. 

EXAMPLES:: 

sage: macaulay2._object_class() 

<class 'sage.interfaces.macaulay2.Macaulay2Element'> 

""" 

return Macaulay2Element 

def console(self): 

""" 

Spawn a new M2 command-line session. 

EXAMPLES:: 

sage: macaulay2.console() # not tested 

Macaulay 2, version 1.1 

with packages: Classic, Core, Elimination, IntegralClosure, LLLBases, Parsing, PrimaryDecomposition, SchurRings, TangentCone 

... 

""" 

macaulay2_console() 

def _left_list_delim(self): 

""" 

Returns the Macaulay2 left delimiter for lists. 

EXAMPLES:: 

sage: macaulay2._left_list_delim() 

'{' 

""" 

return '{' 

def _right_list_delim(self): 

""" 

Returns the Macaulay2 right delimiter for lists. 

EXAMPLES:: 

sage: macaulay2._right_list_delim() 

'}' 

""" 

return '}' 

def _true_symbol(self): 

""" 

Returns the Macaulay2 symbol for True. 

EXAMPLES:: 

sage: macaulay2._true_symbol() 

'true' 

""" 

return 'true' 

def _false_symbol(self): 

""" 

Returns the Macaulay2 symbol for False. 

EXAMPLES:: 

sage: macaulay2._false_symbol() 

'false' 

""" 

return 'false' 

def _equality_symbol(self): 

""" 

Returns the Macaulay2 symbol for equality. 

EXAMPLES:: 

sage: macaulay2._false_symbol() 

'false' 

""" 

return '==' 

def cputime(self, t=None): 

""" 

EXAMPLES:: 

sage: R = macaulay2("QQ[x,y]") # optional - macaulay2 

sage: x,y = R.gens() # optional - macaulay2 

sage: a = (x+y+1)^20 # optional - macaulay2 

sage: macaulay2.cputime() # optional - macaulay2; random 

0.48393700000000001 

""" 

_t = float(self.cpuTime().to_sage()) 

if t: 

return _t - t 

else: 

return _t 

def version(self): 

""" 

Returns the version of Macaulay2. 

EXAMPLES:: 

sage: macaulay2.version() # optional - macaulay2 

(1, 3, 1) 

""" 

s = self.eval("version") 

r = re.compile("VERSION => (.*?)\n") 

s = r.search(s).groups()[0] 

return tuple(int(i) for i in s.split(".")) 

### Constructors 

def ideal(self, *gens): 

""" 

Return the ideal generated by gens. 

INPUT: 

- gens -- list or tuple of Macaulay2 objects (or objects that can be 

made into Macaulay2 objects via evaluation) 

OUTPUT: 

the Macaulay2 ideal generated by the given list of gens 

EXAMPLES:: 

sage: R2 = macaulay2.ring('QQ', '[x, y]'); R2 # optional - macaulay2 

QQ[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, DegreeRank => 1] 

{Lex => 2 } 

{Position => Up } 

sage: I = macaulay2.ideal( ('y^2 - x^3', 'x - y') ); I # optional - macaulay2 

3 2 

ideal (- x + y , x - y) 

sage: J = I^3; J.gb().gens().transpose() # optional - macaulay2 

{-9} | y9-3y8+3y7-y6 | 

{-7} | xy6-2xy5+xy4-y7+2y6-y5 | 

{-5} | x2y3-x2y2-2xy4+2xy3+y5-y4 | 

{-3} | x3-3x2y+3xy2-y3 | 

""" 

if len(gens) == 1 and isinstance(gens[0], (list, tuple)): 

gens = gens[0] 

gens2 = [] 

for g in gens: 

if not isinstance(g, Macaulay2Element): 

gens2.append(self(g)) 

else: 

gens2.append(g) 

return self('ideal {%s}'%(",".join([g.name() for g in gens2]))) 

def ring(self, base_ring='ZZ', vars='[x]', order='Lex'): 

r""" 

Create a Macaulay2 ring. 

INPUT: 

- base_ring -- base ring (see examples below) 

- vars -- a tuple or string that defines the variable names 

- order -- string -- the monomial order (default: 'Lex') 

OUTPUT: 

- a Macaulay2 ring (with base ring ZZ) 

EXAMPLES: 

This is a ring in variables named a through d over the finite field 

of order 7, with graded reverse lex ordering:: 

sage: R1 = macaulay2.ring('ZZ/7', '[a..d]', 'GRevLex'); R1 # optional - macaulay2 

ZZ 

--[a..d, Degrees => {4:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, DegreeRank => 1] 

7 {GRevLex => {4:1} } 

{Position => Up } 

sage: R1.char() # optional - macaulay2 

7 

This is a polynomial ring over the rational numbers:: 

sage: R2 = macaulay2.ring('QQ', '[x, y]'); R2 # optional - macaulay2 

QQ[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, DegreeRank => 1] 

{Lex => 2 } 

{Position => Up } 

""" 

varstr = str(vars)[1:-1] 

if ".." in varstr: 

varstr = "symbol " + varstr[0] + ".." + "symbol " + varstr[-1] 

else: 

varstr = ", ".join(["symbol " + v for v in varstr.split(", ")]) 

return self.new('%s[%s, MonomialSize=>16, MonomialOrder=>%s]'%(base_ring, varstr, order)) 

def help(self, s): 

""" 

EXAMPLES:: 

sage: macaulay2.help("load") # optional - macaulay2 

load -- read Macaulay2 commands 

******************************* 

... 

* "input" -- read Macaulay2 commands and echo 

* "notify" -- whether to notify the user when a file is loaded 

""" 

r = self.eval("help %s" % s) 

end = r.rfind("\n\nDIV") 

if end != -1: 

r = r[:end] 

return AsciiArtString(r) 

def _tab_completion(self): 

""" 

Return a list of tab completions for Macaulay2. 

Returns dynamically built sorted list of commands obtained using 

Macaulay2 "apropos" command. 

Return type: list of strings 

TESTS:: 

sage: names = macaulay2._tab_completion() # optional - macaulay2 

sage: 'ring' in names # optional - macaulay2 

True 

sage: macaulay2.eval("abcabc = 4") # optional - macaulay2 

4 

sage: names = macaulay2._tab_completion() # optional - macaulay2 

sage: "abcabc" in names # optional - macaulay2 

True 

""" 

# Get all the names from Macaulay2 except numbered outputs like 

# o1, o2, etc. and automatic Sage variable names sage0, sage1, etc. 

# It is faster to get it back as a string. 

r = macaulay2.eval(r""" 

toString select( 

apply(apropos "^[[:alnum:]]+$", toString), 

s -> not match("^(o|sage)[0-9]+$", s)) 

""") 

# Now split this string into separate names 

r = sorted(r[1:-1].split(", ")) 

# Macaulay2 sorts things like A, a, B, b, ... 

return r 

def use(self, R): 

""" 

Use the Macaulay2 ring R. 

EXAMPLES:: 

sage: R = macaulay2("QQ[x,y]") # optional - macaulay2 

sage: P = macaulay2("ZZ/7[symbol x, symbol y]") # optional - macaulay2 

sage: macaulay2("x").cls() # optional - macaulay2 

ZZ 

--[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1] 

7 {GRevLex => {2:1} } 

{Position => Up } 

sage: macaulay2.use(R) # optional - macaulay2 

sage: macaulay2("x").cls() # optional - macaulay2 

QQ[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1] 

{GRevLex => {2:1} } 

{Position => Up } 

""" 

R = self(R) 

self.eval("use %s"%R.name()) 

def new_from(self, type, value): 

""" 

Return a new ``Macaulay2Element`` of type ``type`` constructed from 

``value``. 

EXAMPLES:: 

sage: l = macaulay2.new_from("MutableList", [1,2,3]) # optional - macaulay2 

sage: l # optional - macaulay2 

MutableList{...3...} 

sage: list(l) # optional - macaulay2 

[1, 2, 3] 

""" 

type = self(type) 

value = self(value) 

return self.new("new %s from %s"%(type.name(), value.name())) 

@instancedoc 

class Macaulay2Element(ExtraTabCompletion, ExpectElement): 

def _latex_(self): 

""" 

EXAMPLES:: 

sage: m = macaulay2('matrix {{1,2},{3,4}}') # optional - macaulay2 

sage: m # optional - macaulay2 

| 1 2 | 

| 3 4 | 

sage: latex(m) # optional - macaulay2 

\begin{pmatrix}1& {2}\\ {3}& {4}\\ \end{pmatrix} 

""" 

s = self.tex().external_string().strip('"').strip('$').replace('\\\\','\\') 

s = s.replace(r"\bgroup","").replace(r"\egroup","") 

return s 

def __iter__(self): 

""" 

EXAMPLES:: 

sage: l = macaulay2([1,2,3]) # optional - macaulay2 

sage: list(iter(l)) # optional - macaulay2 

[1, 2, 3] 

""" 

for i in range(len(self)): # zero-indexed! 

yield self[i] 

def __str__(self): 

""" 

EXAMPLES:: 

sage: R = macaulay2("QQ[x,y,z]/(x^3-y^3-z^3)") # optional - macaulay2 

sage: x = macaulay2('x') # optional - macaulay2 

sage: y = macaulay2('y') # optional - macaulay2 

sage: print(x+y) # optional - macaulay2 

x + y 

sage: print(macaulay2("QQ[x,y,z]")) # optional - macaulay2 

QQ[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1] 

{GRevLex => {3:1} } 

{Position => Up } 

sage: print(macaulay2("QQ[x,y,z]/(x+y+z)")) # optional - macaulay2 

QQ[x, y, z] 

----------- 

x + y + z 

""" 

P = self._check_valid() 

return P.get(self._name) 

repr = __str__ 

def external_string(self): 

""" 

EXAMPLES:: 

sage: R = macaulay2("QQ[symbol x, symbol y]") # optional - macaulay2 

sage: R.external_string() # optional - macaulay2 

'QQ[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {2:1}, Position => Up}, DegreeRank => 1]' 

""" 

P = self._check_valid() 

code = 'toExternalString(%s)'%self.name() 

X = P.eval(code, strip=True) 

if 'stdio:' in X: 

if 'to external string' in X: 

return P.eval('%s'%self.name()) 

raise RuntimeError("Error evaluating Macaulay2 code.\nIN:%s\nOUT:%s"%(code, X)) 

s = multiple_replace({'\r':'', '\n':' '}, X) 

return s 

def __len__(self): 

""" 

EXAMPLES:: 

sage: l = macaulay2([1,2,3]) # optional - macaulay2 

sage: len(l) # optional - macaulay2 

3 

sage: type(_) # optional - macaulay2 

<... 'int'> 

""" 

self._check_valid() 

return int(self.parent()("#%s"%self.name())) 

def __getitem__(self, n): 

""" 

EXAMPLES:: 

sage: l = macaulay2([1,2,3]) # optional - macaulay2 

sage: l[0] # optional - macaulay2 

1 

""" 

self._check_valid() 

n = self.parent()(n) 

return self.parent().new('%s # %s'%(self.name(), n.name())) 

def __setitem__(self, index, value): 

""" 

EXAMPLES:: 

sage: l = macaulay2.new_from("MutableList", [1,2,3]) # optional - macaulay2 

sage: l[0] = 4 # optional - macaulay2 

sage: list(l) # optional - macaulay2 

[4, 2, 3] 

""" 

P = self.parent() 

index = P(index) 

value = P(value) 

res = P.eval("%s # %s = %s"%(self.name(), index.name(), value.name())) 

if "assignment attempted to element of immutable list" in res: 

raise TypeError("item assignment not supported") 

def __call__(self, x): 

""" 

EXAMPLES:: 

sage: R = macaulay2("QQ[x, y]") # optional - macaulay2 

sage: x,y = R.gens() # optional - macaulay2 

sage: I = macaulay2.ideal(x*y, x+y) # optional - macaulay2 

sage: gb = macaulay2.gb # optional - macaulay2 

sage: gb(I) # optional - macaulay2 

GroebnerBasis[status: done; S-pairs encountered up to degree 1] 

""" 

self._check_valid() 

P = self.parent() 

r = P(x) 

return P('%s %s'%(self.name(), r.name())) 

def __floordiv__(self, x): 

""" 

Quotient of division of self by other. This is denoted //. 

EXAMPLES:: 

sage: R.<x,y> = GF(7)[] 

Now make the M2 version of R, so we can coerce elements of R to M2:: 

sage: macaulay2(R) # optional - macaulay2 

ZZ 

--[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, DegreeRank => 1] 

7 {GRevLex => {2:1} } 

{Position => Up } 

sage: f = (x^3 + 2*y^2*x)^7; f 

x^21 + 2*x^7*y^14 

sage: h = macaulay2(f); h # optional - macaulay2 

21 7 14 

x + 2x y 

sage: f1 = (x^2 + 2*y*x) # optional - macaulay2 

sage: h1 = macaulay2(f1) # optional - macaulay2 

sage: f2 = (x^3 + 2*y*x) # optional - macaulay2 

sage: h2 = macaulay2(f2) # optional - macaulay2 

sage: u = h // [h1,h2] # optional - macaulay2 

sage: h == u[0]*h1 + u[1]*h2 + (h % [h1,h2]) # optional - macaulay2 

True 

""" 

if isinstance(x, (list, tuple)): 

y = self.parent(x) 

z = self.parent().new('%s // matrix{%s}'%(self.name(), y.name())) 

return list(z.entries().flatten()) 

else: 

return self.parent().new('%s // %s'%(self.name(), x.name())) 

def __mod__(self, x): 

""" 

Remainder of division of self by other. This is denoted %. 

EXAMPLES:: 

sage: R.<x,y> = GF(7)[] 

Now make the M2 version of R, so we can coerce elements of R to M2:: 

sage: macaulay2(R) # optional - macaulay2 

ZZ 

--[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, DegreeRank => 1] 

7 {GRevLex => {2:1} } 

{Position => Up } 

sage: f = (x^3 + 2*y^2*x)^7; f # optional - macaulay2 

x^21 + 2*x^7*y^14 

sage: h = macaulay2(f); print(h) # optional - macaulay2 

21 7 14 

x + 2x y 

sage: f1 = (x^2 + 2*y*x) # optional - macaulay2 

sage: h1 = macaulay2(f1) # optional - macaulay2 

sage: f2 = (x^3 + 2*y*x) # optional - macaulay2 

sage: h2 = macaulay2(f2) # optional - macaulay2 

sage: h % [h1,h2] # optional - macaulay2 

-3x*y 

sage: u = h // [h1,h2] # optional - macaulay2 

sage: h == u[0]*h1 + u[1]*h2 + (h % [h1,h2]) # optional - macaulay2 

True 

""" 

if isinstance(x, (list, tuple)): 

y = self.parent(x) 

return self.parent().new('%s %% matrix{%s}'%(self.name(), y.name())) 

if not isinstance(x, Macaulay2Element): 

x = self.parent(x) 

return self.parent().new('%s %% %s'%(self.name(), x.name())) 

def __bool__(self): 

""" 

EXAMPLES:: 

sage: a = macaulay2(0) # optional - macaulay2 

sage: a == 0 # optional - macaulay2 

True 

sage: bool(a) # optional - macaulay2 

False 

""" 

P = self.parent() 

return P.eval('%s == 0'%self.name()) == 'false' 

__nonzero__ = __bool__ 

def sage_polystring(self): 

""" 

If this Macaulay2 element is a polynomial, return a string 

representation of this polynomial that is suitable for 

evaluation in Python. Thus ``*`` is used for multiplication 

and ``**`` for exponentiation. This function is primarily 

used internally. 

EXAMPLES:: 

sage: R = macaulay2.ring('QQ','(x,y)') # optional - macaulay2 

sage: f = macaulay2('x^3 + 3*y^11 + 5') # optional - macaulay2 

sage: print(f) # optional - macaulay2 

3 11 

x + 3y + 5 

sage: f.sage_polystring() # optional - macaulay2 

'x**3+3*y**11+5' 

""" 

return self.external_string().replace('^','**') 

def structure_sheaf(self): 

""" 

EXAMPLES:: 

sage: S = macaulay2('QQ[a..d]') # optional - macaulay2 

sage: R = S/macaulay2('a^3+b^3+c^3+d^3') # optional - macaulay2 

sage: X = R.Proj() # optional - macaulay2 

sage: print(X.structure_sheaf()) # optional - macaulay2 

OO 

sage... 

""" 

return self.parent()('OO_%s'%self.name()) 

def substitute(self, *args, **kwds): 

""" 

Note that we have to override the substitute method so that we get 

the default one from Macaulay2 instead of the one provided by Element. 

EXAMPLES:: 

sage: R = macaulay2("QQ[x]") # optional - macaulay2 

sage: P = macaulay2("ZZ/7[symbol x]") # optional - macaulay2 

sage: x, = R.gens() # optional - macaulay2 

sage: a = x^2 + 1 # optional - macaulay2 

sage: a = a.substitute(P) # optional - macaulay2 

sage: a.to_sage().parent() # optional - macaulay2 

Univariate Polynomial Ring in x over Finite Field of size 7 

""" 

return self.__getattr__("substitute")(*args, **kwds) 

subs = substitute 

def _tab_completion(self): 

""" 

Return a list of tab completions for ``self``. 

Returns dynamically built sorted list of commands obtained using 

Macaulay2 "methods" command. All returned functions can take ``self`` 

as their first argument 

Return type: list of strings 

TESTS:: 

sage: a = macaulay2("QQ[x,y]") # optional - macaulay2 

sage: traits = a._tab_completion() # optional - macaulay2 

sage: "generators" in traits # optional - macaulay2 

True 

""" 

# It is possible, that these are not all possible methods, but 

# there are still plenty and at least there are no definitely 

# wrong ones... 

r = self.parent().eval( 

"""currentClass = class %s; 

total = {}; 

while true do ( 

-- Select methods with first argument of the given class 

r = select(methods currentClass, s -> s_1 === currentClass); 

-- Get their names as strings 

r = apply(r, s -> toString s_0); 

-- Keep only alpha-numeric ones 

r = select(r, s -> match("^[[:alnum:]]+$", s)); 

-- Add to existing ones 

total = total | select(r, s -> not any(total, e -> e == s)); 

if parent currentClass === currentClass then break; 

currentClass = parent currentClass; 

) 

toString total""" % self.name()) 

r = sorted(r[1:-1].split(", ")) 

return r 

def cls(self): 

""" 

Since class is a keyword in Python, we have to use cls to call 

Macaulay2's class. In Macaulay2, class corresponds to Sage's 

notion of parent. 

EXAMPLES:: 

sage: macaulay2(ZZ).cls() # optional - macaulay2 

Ring 

""" 

return self.parent()("class %s"%self.name()) 

########################## 

#Aliases for M2 operators# 

########################## 

def dot(self, x): 

""" 

EXAMPLES:: 

sage: d = macaulay2.new("MutableHashTable") # optional - macaulay2 

sage: d["k"] = 4 # optional - macaulay2 

sage: d.dot("k") # optional - macaulay2 

4 

""" 

parent = self.parent() 

x = parent(x) 

return parent("%s.%s"%(self.name(), x)) 

def _operator(self, opstr, x): 

""" 

Returns the infix binary operation specified by opstr applied 

to self and x. 

EXAMPLES:: 

sage: a = macaulay2("3") # optional - macaulay2 

sage: a._operator("+", a) # optional - macaulay2 

6 

sage: a._operator("*", a) # optional - macaulay2 

9 

""" 

parent = self.parent() 

x = parent(x) 

return parent("%s%s%s"%(self.name(), opstr, x.name())) 

def sharp(self, x): 

""" 

EXAMPLES:: 

sage: a = macaulay2([1,2,3]) # optional - macaulay2 

sage: a.sharp(0) # optional - macaulay2 

1 

""" 

return self._operator("#", x) 

def starstar(self, x): 

""" 

The binary operator ``**`` in Macaulay2 is usually used for tensor 

or Cartesian power. 

EXAMPLES:: 

sage: a = macaulay2([1,2]).set() # optional - macaulay2 

sage: a.starstar(a) # optional - macaulay2 

set {(1, 1), (1, 2), (2, 1), (2, 2)} 

""" 

return self._operator("**", x) 

def underscore(self, x): 

""" 

EXAMPLES:: 

sage: a = macaulay2([1,2,3]) # optional - macaulay2 

sage: a.underscore(0) # optional - macaulay2 

1 

""" 

return self._operator("_", x) 

#################### 

#Conversion to Sage# 

#################### 

def to_sage(self): 

""" 

EXAMPLES:: 

sage: macaulay2(ZZ).to_sage() # optional - macaulay2 

Integer Ring 

sage: macaulay2(QQ).to_sage() # optional - macaulay2 

Rational Field 

sage: macaulay2(2).to_sage() # optional - macaulay2 

2 

sage: macaulay2(1/2).to_sage() # optional - macaulay2 

1/2 

sage: macaulay2(2/1).to_sage() # optional - macaulay2 

2 

sage: _.parent() # optional - macaulay2 

Rational Field 

sage: macaulay2([1,2,3]).to_sage() # optional - macaulay2 

[1, 2, 3] 

sage: m = matrix([[1,2],[3,4]]) 

sage: macaulay2(m).to_sage() # optional - macaulay2 

[1 2] 

[3 4] 

sage: macaulay2(QQ['x,y']).to_sage() # optional - macaulay2 

Multivariate Polynomial Ring in x, y over Rational Field 

sage: macaulay2(QQ['x']).to_sage() # optional - macaulay2 

Univariate Polynomial Ring in x over Rational Field 

sage: macaulay2(GF(7)['x,y']).to_sage() # optional - macaulay2 

Multivariate Polynomial Ring in x, y over Finite Field of size 7 

sage: macaulay2(GF(7)).to_sage() # optional - macaulay2 

Finite Field of size 7 

sage: macaulay2(GF(49, 'a')).to_sage() # optional - macaulay2 

Finite Field in a of size 7^2 

sage: R.<x,y> = QQ[] 

sage: macaulay2(x^2+y^2+1).to_sage() # optional - macaulay2 

x^2 + y^2 + 1 

sage: R = macaulay2("QQ[x,y]") # optional - macaulay2 

sage: I = macaulay2("ideal (x,y)") # optional - macaulay2 

sage: I.to_sage() # optional - macaulay2 

Ideal (x, y) of Multivariate Polynomial Ring in x, y over Rational Field 

sage: X = R/I # optional - macaulay2 

sage: X.to_sage() # optional - macaulay2 

Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x, y) 

sage: R = macaulay2("QQ^2") # optional - macaulay2 

sage: R.to_sage() # optional - macaulay2 

Vector space of dimension 2 over Rational Field 

sage: m = macaulay2('"hello"') # optional - macaulay2 

sage: m.to_sage() # optional - macaulay2 

'hello' 

""" 

repr_str = str(self) 

cls_str = str(self.cls()) 

cls_cls_str = str(self.cls().cls()) 

if repr_str == "ZZ": 

from sage.rings.all import ZZ 

return ZZ 

elif repr_str == "QQ": 

from sage.rings.all import QQ 

return QQ 

if cls_cls_str == "Type": 

if cls_str == "List": 

return [entry.to_sage() for entry in self] 

elif cls_str == "Matrix": 

from sage.matrix.all import matrix 

base_ring = self.ring().to_sage() 

entries = self.entries().to_sage() 

return matrix(base_ring, entries) 

elif cls_str == "Ideal": 

parent = self.ring().to_sage() 

gens = self.gens().entries().flatten().to_sage() 

return parent.ideal(*gens) 

elif cls_str == "QuotientRing": 

#Handle the ZZ/n case 

if "ZZ" in repr_str and "--" in repr_str: 

from sage.rings.all import ZZ, GF 

external_string = self.external_string() 

zz, n = external_string.split("/") 

#Note that n must be prime since it is 

#coming from Macaulay 2 

return GF(ZZ(n)) 

ambient = self.ambient().to_sage() 

ideal = self.ideal().to_sage() 

return ambient.quotient(ideal) 

elif cls_str == "PolynomialRing": 

from sage.rings.all import PolynomialRing 

from sage.rings.polynomial.term_order import inv_macaulay2_name_mapping 

#Get the base ring 

base_ring = self.coefficientRing().to_sage() 

#Get a string list of generators 

gens = str(self.gens())[1:-1] 

# Check that we are dealing with default degrees, i.e. 1's. 

if self.degrees().any("x -> x != {1}").to_sage(): 

raise ValueError("cannot convert Macaulay2 polynomial ring with non-default degrees to Sage") 

#Handle the term order 

external_string = self.external_string() 

order = None 

if "MonomialOrder" not in external_string: 

order = "degrevlex" 

else: 

for order_name in inv_macaulay2_name_mapping: 

if order_name in external_string: 

order = inv_macaulay2_name_mapping[order_name] 

if len(gens) > 1 and order is None: 

raise ValueError("cannot convert Macaulay2's term order to a Sage term order") 

return PolynomialRing(base_ring, order=order, names=gens) 

elif cls_str == "GaloisField": 

from sage.rings.all import ZZ, GF 

gf, n = repr_str.split(" ") 

n = ZZ(n) 

if n.is_prime(): 

return GF(n) 

else: 

gen = str(self.gens())[1:-1] 

return GF(n, gen) 

elif cls_str == "Boolean": 

if repr_str == "true": 

return True 

elif repr_str == "false": 

return False 

elif cls_str == "String": 

return str(repr_str) 

elif cls_str == "Module": 

from sage.modules.all import FreeModule 

if self.isFreeModule().to_sage(): 

ring = self.ring().to_sage() 

rank = self.rank().to_sage() 

return FreeModule(ring, rank) 

else: 

#Handle the integers and rationals separately 

if cls_str == "ZZ": 

from sage.rings.all import ZZ 

return ZZ(repr_str) 

elif cls_str == "QQ": 

from sage.rings.all import QQ 

repr_str = self.external_string() 

if "/" not in repr_str: 

repr_str = repr_str + "/1" 

return QQ(repr_str) 

m2_parent = self.cls() 

parent = m2_parent.to_sage() 

if cls_cls_str == "PolynomialRing": 

from sage.misc.sage_eval import sage_eval 

gens_dict = parent.gens_dict() 

return sage_eval(self.external_string(), gens_dict) 

from sage.misc.sage_eval import sage_eval 

try: 

return sage_eval(repr_str) 

except Exception: 

raise NotImplementedError("cannot convert %s to a Sage object"%repr_str) 

@instancedoc 

class Macaulay2Function(ExpectFunction): 

def _instancedoc_(self): 

""" 

EXAMPLES:: 

sage: print(macaulay2.load.__doc__) # optional - macaulay2 

load -- read Macaulay2 commands 

******************************* 

...