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 https://faculty.math.illinois.edu/Macaulay2/.
No 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.eval(expr)– Evaluation of arbitrary Macaulay2 expressions, with the result returned as a string.macaulay2(expr)– Creation of a Sage object that wraps a Macaulay2 object. This provides a Pythonic interface to Macaulay2. For example, iff = macaulay2(10), thenf.gcd(25)returns the GCD of \(10\) and \(25\) computed using Macaulay2.
EXAMPLES:
sage: 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: R # optional - macaulay2
ZZ
--[x, y, z]
5
sage: x = macaulay2('x') # optional - macaulay2
sage: y = macaulay2('y') # optional - macaulay2
sage: (x+y)^5 # optional - macaulay2
5 5
x + y
sage: parent((x+y)^5) # optional - macaulay2
Macaulay2
The name of the variable to which a Macaulay2 element is assigned internally can be passed as an argument. This is useful for types like polynomial rings which acquire that name in Macaulay2:
sage: R = macaulay2('QQ[x,y,z,w]', 'R') # optional - macaulay2
sage: R # optional - macaulay2
R
sage: f = macaulay2('x^4 + 2*x*y^3 + x*y^2*w + x*y*z*w + x*y*w^2' # optional - macaulay2
....: '+ 2*x*z*w^2 + y^4 + y^3*w + 2*y^2*z*w + z^4 + w^4')
sage: 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)
Use eval() for explicit control over what is sent to the interpreter.
The argument is evaluated in Macaulay2 as is:
sage: macaulay2.eval('compactMatrixForm') # optional - macaulay2
true
sage: macaulay2.eval('compactMatrixForm = false;') # optional - macaulay2
sage: macaulay2.eval('matrix {{1, x^2+y}}') # optional - macaulay2
| 2 |
| 1 x + y |
1 2
Matrix R <--- R
sage: macaulay2.eval('compactMatrixForm = true;') # optional - macaulay2
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.
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class
sage.interfaces.macaulay2.Macaulay2(maxread=None, script_subdirectory=None, logfile=None, server=None, server_tmpdir=None, command=None)¶ Bases:
sage.interfaces.tab_completion.ExtraTabCompletion,sage.interfaces.expect.ExpectInterface to the Macaulay2 interpreter.
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clear(var)¶ Clear the variable named
var.The interface automatically clears Macaulay2 elements when they fall out of use, so calling this method is usually not necessary.
EXAMPLES:
sage: macaulay2.eval('R = QQ[x,y];') # optional - macaulay2 sage: macaulay2.eval('net class R') # optional - macaulay2 PolynomialRing sage: macaulay2.clear('R') # optional - macaulay2 sage: macaulay2.eval('net class R') # optional - macaulay2 Symbol
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console()¶ 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 ...
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cputime(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
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eval(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
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get(var)¶ Get the value of the variable
var.INPUT:
var- string; the name of the variable in Macaulay2
OUTPUT: a string of the textual representation of the variable in Macaulay2
EXAMPLES:
sage: macaulay2.set("a", "2") # optional - macaulay2 sage: macaulay2.get("a") # optional - macaulay2 2
Note that the following syntax is used to obtain a
Macaulay2Elementinstead:sage: a = macaulay2('2'); a # optional - macaulay2 2 sage: type(a) # optional - macaulay2 <class 'sage.interfaces.macaulay2.Macaulay2Element'>
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help(s)¶ EXAMPLES:
sage: macaulay2.help("load") # optional - macaulay2 - 1st call might be chatty... ... sage: macaulay2.help("load") # optional - macaulay2 load... ****... ... * "input" -- read Macaulay2 commands and echo * "notify" -- whether to notify the user when a file is loaded
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ideal(*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] 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 |
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new_from(type, value)¶ Return a new
Macaulay2Elementof typetypeconstructed fromvalue.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]
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options(*get_value, **set_value)¶ Global options for Macaulay2 elements.
OPTIONS:
after_print– (default:False) append AfterPrint type information to textual representations
EXAMPLES:
sage: macaulay2.options.after_print = True # optional - macaulay2 sage: A = macaulay2(matrix([[1, 2], [3, 6]])); A # optional - macaulay2 | 1 2 | | 3 6 | 2 2 Matrix ZZ <--- ZZ sage: A.kernel() # optional - macaulay2 image | 2 | | -1 | 2 ZZ-module, submodule of ZZ sage: macaulay2.options.after_print = False # optional - macaulay2
See
GlobalOptionsfor more features of these options.
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restart()¶ Restart Macaulay2 interpreter.
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ring(base_ring='ZZ', vars='[x]', order='Lex')¶ Create a Macaulay2 polynomial ring.
INPUT:
base_ring– base ring (see examples below)vars– a tuple or string that defines the variable namesorder– string (default: ‘Lex’); the monomial order
OUTPUT: a Macaulay2 ring
EXAMPLES:
This is a ring in variables named
athroughdover the finite field of order 7, with graded reverse lex ordering:sage: R1 = macaulay2.ring('ZZ/7', '[a..d]', 'GRevLex') # optional - macaulay2 sage: R1.describe() # optional - macaulay2 ZZ --[a..d, Degrees => {4:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, 7 {GRevLex => {4:1} } {Position => Up } -------------------------------------------------------------------------------- DegreeRank => 1] sage: R1.char() # optional - macaulay2 7
This is a polynomial ring over the rational numbers:
sage: R2 = macaulay2.ring('QQ', '[x, y]') # optional - macaulay2 sage: R2.describe() # optional - macaulay2 QQ[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, {Lex => 2 } {Position => Up } -------------------------------------------------------------------------------- DegreeRank => 1]
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set(var, value)¶ Set the variable
varto the given value.INPUT:
var- string; the name of the variable in Macaulay2value- a string to evaluate
EXAMPLES:
sage: macaulay2.set("a", "1+1") # optional - macaulay2 sage: macaulay2.get("a") # optional - macaulay2 2
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set_seed(seed=None)¶ Set the seed for Macaulay2 interpreter.
INPUT:
seed– number (default:None). IfNone, it is set to a random number.
OUTPUT: the new seed
EXAMPLES:
sage: m = Macaulay2() # optional - macaulay2 sage: m.set_seed(123456) # optional - macaulay2 123456 sage: [m.random(100) for _ in range(11)] # optional - macaulay2 [8, 29, 5, 22, 4, 32, 35, 57, 3, 95, 36]
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use(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()._operator('===', P) # optional - macaulay2 true sage: macaulay2.use(R) # optional - macaulay2 sage: macaulay2("x").cls()._operator('===', R) # optional - macaulay2 true
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version()¶ Returns the version of Macaulay2.
EXAMPLES:
sage: macaulay2.version() # optional - macaulay2 (1, 1...
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class
sage.interfaces.macaulay2.Macaulay2Element(parent, value, is_name=False, name=None)¶ Bases:
sage.interfaces.tab_completion.ExtraTabCompletion,sage.interfaces.expect.ExpectElementInstances of this class represent objects in Macaulay2.
Using the method
sage()we can translate some of them to SageMath objects:-
_sage_()¶ EXAMPLES:
sage: macaulay2(ZZ).sage() # optional - macaulay2, indirect doctest Integer Ring sage: macaulay2(QQ).sage() # optional - macaulay2 Rational Field sage: macaulay2(2).sage() # optional - macaulay2 2 sage: macaulay2(1/2).sage() # optional - macaulay2 1/2 sage: macaulay2(2/1).sage() # optional - macaulay2 2 sage: _.parent() # optional - macaulay2 Rational Field sage: macaulay2([1,2,3]).sage() # optional - macaulay2 [1, 2, 3] sage: m = matrix([[1,2],[3,4]]) sage: macaulay2(m).sage() # optional - macaulay2 [1 2] [3 4] sage: D = macaulay2('hashTable {4 => 1, 2 => 3}') # optional - macaulay2 sage: D.pairs() # optional - macaulay2 {(4, 1), (2, 3)} sage: D.sage() == {4: 1, 2: 3} # optional - macaulay2 True sage: macaulay2(QQ['x,y']).sage() # optional - macaulay2 Multivariate Polynomial Ring in x, y over Rational Field sage: macaulay2(QQ['x']).sage() # optional - macaulay2 Univariate Polynomial Ring in x over Rational Field sage: macaulay2(GF(7)['x,y']).sage() # optional - macaulay2 Multivariate Polynomial Ring in x, y over Finite Field of size 7 sage: macaulay2(GF(7)).sage() # optional - macaulay2 Finite Field of size 7 sage: macaulay2(GF(49, 'a')).sage() # optional - macaulay2 Finite Field in a of size 7^2 sage: R.<x,y> = QQ[] sage: macaulay2(x^2+y^2+1).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.sage() # optional - macaulay2 Ideal (x, y) of Multivariate Polynomial Ring in x, y over Rational Field sage: macaulay2("x = symbol x") # optional - macaulay2 x sage: macaulay2("QQ[x_0..x_25]").sage() # optional - macaulay2 Multivariate Polynomial Ring in x_0, x_1,..., x_25 over Rational Field sage: S = ZZ['x,y'].quotient('x^2-y') sage: macaulay2(S).sage() == S # optional - macaulay2 True sage: S = GF(101)['x,y'].quotient('x^2-y') sage: macaulay2(S).sage() == S # optional - macaulay2 True sage: R = GF(13)['a,b']['c,d'] sage: macaulay2(R).sage() == R # optional - macaulay2 True sage: macaulay2('a^2 + c').sage() == R('a^2 + c') # optional - macaulay2 True sage: macaulay2.substitute('a', R).sage().parent() is R # optional - macaulay2 True sage: R = macaulay2("QQ^2") # optional - macaulay2 sage: R.sage() # optional - macaulay2 Vector space of dimension 2 over Rational Field sage: macaulay2("vector {4_QQ, 2}").sage() # optional - macaulay2 (4, 2) sage: _.parent() # optional - macaulay2 Vector space of dimension 2 over Rational Field sage: m = macaulay2('"hello"') # optional - macaulay2 sage: m.sage() # optional - macaulay2 'hello' sage: gg = macaulay2.needsPackage('"Graphs"') # optional - macaulay2 sage: g = macaulay2.barbellGraph(3) # optional - macaulay2 sage: g.sage() # optional - macaulay2 Graph on 6 vertices sage: g.sage().edges(labels=False) # optional - macaulay2 [(0, 1), (0, 2), (1, 2), (2, 3), (3, 4), (3, 5), (4, 5)] sage: d = 'digraph ({{1,2},{2,1},{3,1}}, EntryMode => "edges")' sage: g = macaulay2(d) # optional - macaulay2 sage: g.sage() # optional - macaulay2 Digraph on 3 vertices sage: g.sage().edges(labels=False) # optional - macaulay2 [(1, 2), (2, 1), (3, 1)]
Chain complexes and maps of chain complexes can be converted:
sage: R = ZZ['a,b,c'] sage: C = macaulay2(ideal(R.gens())).resolution() # optional - macaulay2 sage: ascii_art(C.sage()) # optional - macaulay2 [-b 0 -c] [ c] [ a -c 0] [ a] [a b c] [ 0 b a] [-b] 0 <-- C_0 <-------- C_1 <----------- C_2 <----- C_3 <-- 0 sage: F = C.dot('dd') # optional - macaulay2 sage: G = F.sage() # optional - macaulay2 sage: G.in_degree(2) # optional - macaulay2 [-b 0 -c] [ a -c 0] [ 0 b a] sage: F.underscore(2).sage() == G.in_degree(2) # optional - macaulay2 True sage: (F^2).sage() # optional - macaulay2 Chain complex morphism: From: Chain complex with at most 4 nonzero terms over Multivariate Polynomial Ring in a, b, c over Integer Ring To: Chain complex with at most 4 nonzero terms over Multivariate Polynomial Ring in a, b, c over Integer Ring
Quotient rings in Macaulay2 inherit variable names from the ambient ring, so we mimic this behaviour in Sage:
sage: R = macaulay2("ZZ/7[x,y]") # optional - macaulay2 sage: I = macaulay2("ideal (x^3 - y^2)") # optional - macaulay2 sage: (R/I).gens() # optional - macaulay2 {x, y} sage: (R/I).sage().gens() # optional - macaulay2 (x, y)
Elements of quotient rings:
sage: x, y = (R/I).gens() # optional - macaulay2 sage: f = ((x^3 + 2*y^2*x)^7).sage(); f # optional - macaulay2 2*x*y^18 + y^14 sage: f.parent() # optional - macaulay2 Quotient of Multivariate Polynomial Ring in x, y over Finite Field of size 7 by the ideal (x^3 - y^2)
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after_print_text()¶ Obtain type information for this Macaulay2 element.
This is the text that is displayed using
AfterPrintin a Macaulay2 interpreter.Macaulay2 by default includes this information in the output. In Sage, this behavior can optionally be enabled by setting the option
after_printinMacaulay2.options.EXAMPLES:
sage: B = macaulay2(matrix([[1, 2], [3, 6]])).kernel(); B # optional - macaulay2 image | 2 | | -1 | sage: B.after_print_text() # optional - macaulay2 2 ZZ-module, submodule of ZZ
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cls()¶ 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
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dot(x)¶ EXAMPLES:
sage: d = macaulay2.new("MutableHashTable") # optional - macaulay2 sage: d["k"] = 4 # optional - macaulay2 sage: d.dot("k") # optional - macaulay2 4
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external_string()¶ EXAMPLES:
sage: R = macaulay2("QQ[symbol x, symbol y]") # optional - macaulay2 sage: R.external_string() # optional - macaulay2 'QQ(monoid[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {2:1}, Position => Up}, DegreeRank => 1])'
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name(new_name=None)¶ Get or change the name of this Macaulay2 element.
INPUT:
new_name– string (default:None). IfNone, return the name of this element; else return a new object identical toselfwhose name isnew_name.
Note that this can overwrite existing variables in the system.
EXAMPLES:
sage: S = macaulay2(QQ['x,y']) # optional - macaulay2 sage: S.name() # optional - macaulay2 'sage...' sage: R = S.name("R") # optional - macaulay2 sage: R.name() # optional - macaulay2 'R' sage: R.vars().cokernel().resolution() # optional - macaulay2 1 2 1 R <-- R <-- R <-- 0 0 1 2 3
The name can also be given at definition:
sage: A = macaulay2(ZZ['x,y,z'], name='A') # optional - macaulay2 sage: A.name() # optional - macaulay2 'A' sage: A^1 # optional - macaulay2 1 A
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sage_polystring()¶ 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'
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sharp(x)¶ EXAMPLES:
sage: a = macaulay2([1,2,3]) # optional - macaulay2 sage: a.sharp(0) # optional - macaulay2 1
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starstar(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)}
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structure_sheaf()¶ 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().name('X') # optional - macaulay2 sage: X.structure_sheaf() # optional - macaulay2 doctest:...: DeprecationWarning: The function `structure_sheaf` is deprecated. Use `self.sheaf()` instead. See https://trac.sagemath.org/27848 for details. OO X sage: X.sheaf() # optional - macaulay2 OO X
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subs(*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.sage().parent() # optional - macaulay2 Univariate Polynomial Ring in x over Finite Field of size 7
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substitute(*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.sage().parent() # optional - macaulay2 Univariate Polynomial Ring in x over Finite Field of size 7
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to_sage(*args, **kwds)¶ Deprecated: Use
sage()instead. See trac ticket #27848 for details.
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underscore(x)¶ EXAMPLES:
sage: a = macaulay2([1,2,3]) # optional - macaulay2 sage: a.underscore(0) # optional - macaulay2 1
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class
sage.interfaces.macaulay2.Macaulay2Function(parent, name)¶
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class
sage.interfaces.macaulay2.Macaulay2FunctionElement(obj, name)¶
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sage.interfaces.macaulay2.is_Macaulay2Element(x)¶ EXAMPLES:
sage: from sage.interfaces.macaulay2 import is_Macaulay2Element sage: is_Macaulay2Element(2) # optional - macaulay2 False sage: is_Macaulay2Element(macaulay2(2)) # optional - macaulay2 True
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sage.interfaces.macaulay2.macaulay2_console()¶ 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 ...
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sage.interfaces.macaulay2.reduce_load_macaulay2()¶ Used for reconstructing a copy of the Macaulay2 interpreter from a pickle.
EXAMPLES:
sage: from sage.interfaces.macaulay2 import reduce_load_macaulay2 sage: reduce_load_macaulay2() Macaulay2
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sage.interfaces.macaulay2.remove_output_labels(s)¶ 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
sconsists of several outputs and their labels 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'