Trac #29629: Ore polynomials

We implement general univariate Ore polynomials (allowing for
derivations and twisted derivations)

URL: https://trac.sagemath.org/29629
Reported by: caruso
Ticket author(s): Xavier Caruso
Reviewer(s): Travis Scrimshaw

src/doc/en/reference/polynomial_rings/index.rst

@@ -29,15 +29,18 @@ Multivariate Polynomials
    invariant_theory
    polynomial_rings_toy_implementations
 
-Skew Polynomials
-----------------
+Ore Polynomials
+---------------
 
 .. toctree::
    :maxdepth: 2
 
-   sage/rings/polynomial/skew_polynomial_element
+   sage/rings/polynomial/ore_polynomial_ring
+   sage/rings/polynomial/ore_polynomial_element
    sage/rings/polynomial/skew_polynomial_ring
+   sage/rings/polynomial/skew_polynomial_element
    sage/rings/polynomial/skew_polynomial_finite_order
+   sage/rings/polynomial/skew_polynomial_finite_field
 
 Rational Functions
 ------------------

src/module_list.py

@@ -1110,6 +1110,9 @@
     Extension('sage.rings.polynomial.symmetric_reduction',
               sources = ['sage/rings/polynomial/symmetric_reduction.pyx']),
 
+    Extension('sage.rings.polynomial.ore_polynomial_element',
+              sources = ['sage/rings/polynomial/ore_polynomial_element.pyx']),
+
     Extension('sage.rings.polynomial.skew_polynomial_element',
               sources = ['sage/rings/polynomial/skew_polynomial_element.pyx']),
 

src/sage/categories/rings.py

@@ -926,12 +926,17 @@ def __getitem__(self, arg):
                 sage: GF(17)['a']['b']
                 Univariate Polynomial Ring in b over Univariate Polynomial Ring in a over Finite Field of size 17
 
-            We can create skew polynomial rings::
+            We can create Ore polynomial rings::
 
                 sage: k.<t> = GF(5^3)
                 sage: Frob = k.frobenius_endomorphism()
-                sage: k['x',Frob]
-                Skew Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
+                sage: k['x', Frob]
+                Ore Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
+
+                sage: R.<t> = QQ[]
+                sage: der = R.derivation()
+                sage: R['d', der]
+                Ore Polynomial Ring in d over Univariate Polynomial Ring in t over Rational Field twisted by d/dt
 
             We can also create power series rings by using double brackets::
 
@@ -1080,9 +1085,10 @@ def normalize_arg(arg):
 
             if isinstance(arg, tuple):
                 from sage.categories.morphism import Morphism
-                if len(arg) == 2 and isinstance(arg[1], Morphism):
-                   from sage.rings.polynomial.skew_polynomial_ring import SkewPolynomialRing
-                   return SkewPolynomialRing(self, arg[1], names=arg[0])
+                from sage.rings.derivation import RingDerivation
+                if len(arg) == 2 and isinstance(arg[1], (Morphism, RingDerivation)):
+                    from sage.rings.polynomial.ore_polynomial_ring import OrePolynomialRing
+                    return OrePolynomialRing(self, arg[1], names=arg[0])
 
             # 2. Otherwise, if all specified elements are algebraic, try to
             #    return an algebraic extension

src/sage/rings/derivation.py

@@ -203,7 +203,9 @@
 from sage.categories.lie_algebras import LieAlgebras
 
 from sage.categories.map import Map
-from sage.categories.all import Rings
+from sage.categories.rings import Rings
+
+from sage.misc.latex import latex
 
 
 class RingDerivationModule(Module, UniqueRepresentation):
@@ -473,7 +475,6 @@ def _repr_(self):
             sage: R.derivation_module(twist=theta)
             Module of twisted derivations over Multivariate Polynomial Ring in x, y
              over Integer Ring (twisting morphism: x |--> y, y |--> x)
-
         """
         t = ""
         if self._twist is None:
@@ -850,7 +851,6 @@ def _repr_(self):
             d/dx
             sage: R.derivation(y)
             d/dy
-
         """
         parent = self.parent()
         try:
@@ -884,6 +884,56 @@ def _repr_(self):
         else:
             return s
 
+    def _latex_(self):
+        r"""
+        Return a LaTeX representation of this derivation.
+
+        EXAMPLES::
+
+            sage: R.<x,y> = ZZ[]
+            sage: ddx = R.derivation(x)
+            sage: ddy = R.derivation(y)
+        
+            sage: latex(ddx)
+            \frac{d}{dx}
+            sage: latex(ddy)
+            \frac{d}{dy}
+            sage: latex(ddx + ddy)
+            \frac{d}{dx} + \frac{d}{dy}
+        """
+        parent = self.parent()
+        try:
+            dual_basis = parent.dual_basis()
+        except NotImplementedError:
+            return "\\text{A derivation on } %s" % latex(parent.domain())
+        coeffs = self.list()
+        s = ""
+        for i in range(len(dual_basis)):
+            c = coeffs[i]
+            sc = str(c)
+            if sc == "0":
+                continue
+            ddx = "\\frac{d}{d%s}" % latex(dual_basis[i])
+            if sc == "1":
+                s += " + " + ddx
+            elif sc == "-1":
+                s += " - " + ddx
+            elif c._is_atomic() and sc[0] != "-":
+                s += " + %s %s" % (sc, ddx)
+            elif (-c)._is_atomic():
+                s += " - %s %s" % (-c, ddx)
+            else:
+                s += " + \\left(%s\\right) %s" % (sc, ddx)
+        if s[:3] == " + ":
+            return s[3:]
+        elif s[:3] == " - ":
+            return "-" + s[3:]
+        elif s == "":
+            return "0"
+        else:
+            return s
+
+
     def list(self):
         """
         Return the list of coefficient of this derivation
@@ -1268,7 +1318,18 @@ def _repr_(self):
             sage: M = ZZ.derivation_module()
             sage: M()
             0
+        """
+        return "0"
+
+    def _latex_(self):
+        """
+        Return a string representation of this derivation.
 
+        EXAMPLES::
+
+            sage: M = ZZ.derivation_module()
+            sage: latex(M())
+            0
         """
         return "0"
 
@@ -1911,7 +1972,6 @@ def _repr_(self):
             sage: theta = R.hom([y,x])
             sage: R.derivation(1, twist=theta)
             [x |--> y, y |--> x] - id
-
         """
         scalar = self._scalar
         sc = str(scalar)
@@ -1942,6 +2002,47 @@ def _repr_(self):
             s = "(%s)*" % sc
         return "%s(%s - %s)" % (s, stwi, sdef)
 
+    def _latex_(self):
+        r"""
+        Return a LaTeX representation of this derivation.
+
+        EXAMPLES::
+
+            sage: k.<a> = GF(5^3)
+            sage: Frob = k.frobenius_endomorphism()
+            sage: der = k.derivation(a+1, twist=Frob)
+            sage: latex(der)
+            \left(a + 1\right)  \left(\left[a \mapsto a^{5}\right] - \text{id}\right)
+        """
+        scalar = self._scalar
+        sc = str(scalar)
+        if sc == "0":
+            return "0"
+        defining_morphism = self.parent().defining_morphism()
+        twisting_morphism = self.parent().twisting_morphism()
+        try:
+            if defining_morphism.is_identity():
+                sdef = "\\text{id}"
+            else:
+                sdef = "\\left[%s\\right]" % latex(defining_morphism)
+        except (AttributeError, NotImplementedError):
+            sdef = "\\text{defining morphism}"
+        try:
+            stwi = "\\left[%s\\right]" % latex(twisting_morphism)
+        except AttributeError:
+            stwi = "\\text{twisting morphism}"
+        if sc == "1":
+            return "%s - %s" % (stwi, sdef)
+        elif sc == "-1":
+            s = "-"
+        elif scalar._is_atomic():
+            s = "%s " % sc
+        elif (-scalar)._is_atomic():
+            s = "-%s " % (-scalar)
+        else:
+            s = "\\left(%s\\right) " % sc
+        return "%s \\left(%s - %s\\right)" % (s, stwi, sdef)
+
     def _add_(self, other):
         """
         Return the sum of this derivation and ``other``.
@@ -2173,4 +2274,3 @@ def _richcmp_(self, other, op):
             else:
                 return True
         return NotImplemented
-

src/sage/rings/polynomial/all.py

@@ -45,8 +45,9 @@
 # Infinite Polynomial Rings
 from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing
 
-# Skew Polynomial Rings
-from sage.rings.polynomial.skew_polynomial_ring import SkewPolynomialRing
+# Ore Polynomial Rings
+lazy_import('sage.rings.polynomial.ore_polynomial_ring', 'OrePolynomialRing')
+SkewPolynomialRing = OrePolynomialRing
 
 # Evaluation of cyclotomic polynomials
 from sage.rings.polynomial.cyclotomic import cyclotomic_value

src/sage/rings/polynomial/ore_polynomial_element.pxd

@@ -0,0 +1,46 @@
+from sage.structure.element cimport AlgebraElement
+from sage.structure.parent cimport Parent
+from sage.rings.morphism cimport Morphism
+from sage.structure.element cimport RingElement
+from sage.rings.polynomial.polynomial_element cimport Polynomial_generic_dense
+
+cdef class OrePolynomial(AlgebraElement):
+    cdef _is_gen
+
+    cdef long _hash_c(self)
+    cdef OrePolynomial _new_c(self, list coeffs, Parent P, char check=*)
+    cpdef OrePolynomial _new_constant_poly(self, RingElement a, Parent P, char check=*)
+    cpdef _neg_(self)
+    cpdef _floordiv_(self, right)
+    cpdef _mod_(self, right)
+
+    cpdef bint is_zero(self)
+    cpdef bint is_one(self)
+
+    cdef _left_quo_rem(self, OrePolynomial other)
+    cdef _right_quo_rem(self, OrePolynomial other)
+    cdef OrePolynomial _left_lcm_cofactor(self, OrePolynomial other)
+    cdef OrePolynomial _right_lcm_cofactor(self, OrePolynomial other)
+
+    # Abstract methods
+    cpdef int degree(self)
+    cpdef list coefficients(self, sparse=*)
+
+
+cdef void lmul_gen(list A, Morphism m, d)
+
+cdef class OrePolynomial_generic_dense(OrePolynomial):
+    cdef list _coeffs
+
+    cdef void __normalize(self)
+    cpdef _add_(self, other)
+    cdef list _mul_list(self, list A)
+    cpdef _mul_(self, other)
+
+    cpdef dict dict(self)
+    cpdef list list(self, bint copy=*)
+
+
+cdef class OrePolynomialBaseringInjection(Morphism):
+    cdef RingElement _an_element
+    cdef object _new_constant_poly_