Trac #13630: implement padic (x)gcd for fixed mod polynomials

src/sage/rings/padics/generic_nodes.py

@@ -8,7 +8,18 @@
 AUTHORS:
 
 - David Roe
+
+- Julian Rueth (2012-09-05): (x)gcd for polynomials
 """
+#*****************************************************************************
+#       Copyright (C) 2009 David Roe <roed@math.harvard.edu>
+#                     2012 Julian Rueth <julian.rueth@fsfe.org>
+#
+#  Distributed under the terms of the GNU General Public License (GPL)
+#  as published by the Free Software Foundation; either version 2 of
+#  the License, or (at your option) any later version.
+#                  http://www.gnu.org/licenses/
+#*****************************************************************************
 
 from sage.rings.padics.local_generic import LocalGeneric
 from sage.rings.padics.padic_generic import pAdicGeneric
@@ -229,7 +240,176 @@ class pAdicFieldGeneric(pAdicGeneric, Field):
     #    raise NotImplementedError
 
 class pAdicFixedModRingGeneric(pAdicRingGeneric, FixedModGeneric):
-    pass
+    def _gcd_univariate_polynomial(self, f, g):
+        """
+        Compute a greatest common divisor of the polynomials ``f`` and ``g``.
+
+        This is a helper method for
+        :meth:`sage.rings.polynomial.polynomial_element.Polynomial.gcd`. Its
+        implementation relies on
+        :meth:`sage.rings.padics.padic_generic.pAdicGeneric._gcd_univariate_polynomial_fixed`
+        which should be consulted for further details and examples.
+
+        INPUT:
+
+            - ``f``, ``g`` -- two polynomials defined over ``self``.
+
+        OUTPUT:
+
+        A polynomial defined over ``self``, and the precision to which this
+        result is accurate (see
+        :meth:`sage.rings.padics.padic_generic.pAdicGeneric.__xgcd_univariate_polynomial_fixed`
+        for the precise meaning of this accuracy.)
+
+        AUTHORS:
+
+        - Julian Rueth (2012-09-05): initial version
+
+        EXAMPLES::
+
+            sage: R.<t> = ZpFM(3,20)[]
+            sage: (t + 1).gcd( (t - 1) * (t + 1) )
+            ((1 + O(3^20))*t + (1 + O(3^20)), 20)
+            sage: (t^3).gcd( t^5 )
+            ((1 + O(3^20))*t^3, 20)
+
+        Also works over extensions::
+
+            sage: K = ZpFM(3,20)
+            sage: R.<a> = K[]
+            sage: L.<a> = K.extension( a^2 - 3 ) # Eisenstein extension
+            sage: R.<t> = L[]
+            sage: a0,a1,a2 = 12345678+a,90123456-a,78901234*a
+            sage: f = (t - a0) * (t - a1)^2 * (t - a2)^3
+            sage: g,prec = f.gcd(f.derivative())
+            sage: h = ((t - a1) * (t - a2)^2) # the correct result
+            sage: g.degree() == h.degree()
+            True
+            sage: all([(c1-c2).is_zero(prec) for c1,c2 in zip(list(g),list(h))]) # g and h are equal mod a^prec
+            True
+
+            sage: R.<a> = K[]
+            sage: L.<a> = K.extension( a^2 - 2 ) # unramified extension
+            sage: R.<t> = L[]
+            sage: a0,a1,a2 = 12345678+a,90123456-a,78901234*a
+            sage: f = (t - a0) * (t - a1)^2 * (t - a2)^3
+            sage: g,prec = f.gcd(f.derivative())
+            sage: h = ((t - a1) * (t - a2)^2) # the correct result
+            sage: g.degree() == h.degree()
+            True
+            sage: all([(c1-c2).is_zero(prec) for c1,c2 in zip(list(g),list(h))]) # g and h are equal mod p^prec
+            True
+
+        TESTS:
+
+        Check that the examples from :trac:`13439` work::
+
+            sage: R.<t> = ZpFM(3,3)[]
+            sage: f = 3*t + 7
+            sage: g = 5*t + 9
+            sage: f.gcd(f*g)
+            ((3 + O(3^3))*t + (1 + 2*3 + O(3^3)), 2)
+
+            sage: R.<t> = ZpFM(3,20)[]
+            sage: f = 729*490473657*t + 257392844
+            sage: g = 225227399*t - 59049*8669753175
+            sage: h,prec = f.gcd(f*g)
+            sage: h.degree() == f.degree()
+            True
+            sage: h *= f.leading_coefficient().unit_part()
+            sage: all([(c1-c2).is_zero(prec) for c1,c2 in zip(list(f),list(h))]) # f and h are equal mod p^prec
+            True
+
+        """
+        return self._gcd_univariate_polynomial_fixed(f, g)
+
+    def _xgcd_univariate_polynomial(self, f, g):
+        """
+        Compute an extended greatest common divisor of the polynomials ``f``
+        and ``g``.
+
+        This is a helper method for
+        :meth:`sage.rings.polynomial.polynomial_element.Polynomial.xgcd`. Its
+        implementation relies on
+        :meth:`sage.rings.padics.padic_generic.pAdicGeneric._xgcd_univariate_polynomial_fixed`
+        which should be consulted for further details and examples.
+
+        INPUT:
+
+            - ``f``, ``g`` -- two polynomials defined over ``self``.
+
+        OUTPUT:
+
+        A tuple ``r,prec,s,t`` which satisfies ``r = s*f + t*g`` when reduced
+        to precision ``prec``.  (see
+        :meth:`sage.rings.padics.padic_generic.pAdicGeneric.__xgcd_univariate_polynomial_fixed`
+        for the precise meaning of the precision ``prec``.)
+
+        AUTHORS:
+
+        - Julian Rueth (2012-10-22): initial version
+
+        EXAMPLES::
+
+            sage: R.<t> = ZpFM(3,20)[]
+            sage: (t + 1).xgcd( (t - 1) * (t + 1) )
+            ((1 + O(3^20))*t + (1 + O(3^20)), 20, (1 + O(3^20)), 0)
+            sage: (t^3).xgcd( t^5 )
+            ((1 + O(3^20))*t^3, 20, (1 + O(3^20)), 0)
+
+        Also works over extensions::
+
+            sage: K = ZpFM(3,20)
+            sage: R.<a> = K[]
+            sage: L.<a> = K.extension( a^2 - 3 ) # Eisenstein extension
+            sage: R.<t> = L[]
+            sage: a0,a1,a2 = 12345678+a,90123456-a,78901234*a
+            sage: f = (t - a0) * (t - a1)^2 * (t - a2)^3
+            sage: g,prec,u,v = f.xgcd(f.derivative())
+            sage: h = (t - a1) * (t - a2)^2 # the correct result
+            sage: g.degree() == h.degree()
+            True
+            sage: h *= g.leading_coefficient()
+            sage: all([(c1-c2).is_zero(prec) for c1,c2 in zip(list(g),list(h))]) # g and h are equal mod a^prec
+            True
+            sage: all([(c1-c2).is_zero(prec) for c1,c2 in zip(list(g),list(u*f+v*f.derivative()))]) # the equation g = u*f + v*f.derivative() is satisfied mod a^prec
+            True
+
+            sage: R.<a> = K[]
+            sage: L.<a> = K.extension( a^2 - 2 ) # unramified extension
+            sage: R.<t> = L[]
+            sage: a0,a1,a2 = 12345678+a,90123456-a,78901234*a
+            sage: f = (t - a0) * (t - a1)^2 * (t - a2)^3
+            sage: g,prec,u,v = f.xgcd(f.derivative())
+            sage: h = (t - a1) * (t - a2)^2 # the correct result
+            sage: g.degree() == h.degree()
+            True
+            sage: h *= g.leading_coefficient()
+            sage: all([(c1-c2).is_zero(prec) for c1,c2 in zip(list(g),list(h))]) # g and h are equal mod a^prec
+            True
+            sage: all([(c1-c2).is_zero(prec) for c1,c2 in zip(list(g),list(u*f+v*f.derivative()))]) # the equation g = u*f + v*f.derivative() is satisfied mod a^prec
+            True
+
+        TESTS:
+
+        Check that the examples from :trac:`13439` work::
+
+            sage: R.<t> = ZpFM(3,3)[]
+            sage: f = 3*t + 7
+            sage: g = 5*t + 9
+            sage: f.xgcd(f*g)
+            ((3 + O(3^3))*t + (1 + 2*3 + O(3^3)), 2, (1 + O(3^3)), 0)
+
+            sage: R.<t> = ZpFM(3,20)[]
+            sage: f = 729*490473657*t + 257392844
+            sage: g = 225227399*t - 59049*8669753175
+            sage: h,_,_,_ = f.xgcd(f*g)
+            sage: h.degree() == f.degree()
+            True
+
+        """
+        return self._xgcd_univariate_polynomial_fixed(f, g)
+
 class pAdicCappedAbsoluteRingGeneric(pAdicRingGeneric, CappedAbsoluteGeneric):
     pass
 class pAdicCappedRelativeRingGeneric(pAdicRingGeneric, CappedRelativeRingGeneric):