\hSSKJr SSKJrJrJr SSKJr SSKJ r J r J r J r J r SSKJrJr SSKJrJr SSKJrJr SrS S S .S jrg ))combinations_with_replacement)symbolsAddDummy)Rational)cancelComputationFailedparallel_poly_from_exprreducedPoly)Monomial monomial_div) DomainErrorPolificationFailed)debugdebugfc[U5R5up[X/SSS9up4[ U6[XB- 5-$![a X- s$f=f)z Put an expression over a common denominator, cancel and reduce. Examples ======== >>> from sympy import ratsimp >>> from sympy.abc import x, y >>> ratsimp(1/x + 1/y) (x + y)/(x*y) TF)fieldexpand)ras_numer_denomr r r)exprfgQrs N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/simplify/ratsimp.pyratsimpr s^ $< & & (DAq#T%8 7VAC[   s sAAATF)quick polynomialc ^^^^^^^^SSKJm [SU5 [U5R 5upg[ Xg/T-/UQ70UD6unmTRn U R(aU R5TlO[SU -5eUSSV s/sHoRTR5PM sn m[5mUUU4SjmSUUUUUUU4Sjjm[UTTRTRS 9S n[UTTRTRS 9S nU(aXg- R5$T"[!UTRTRS 9[!UTRTRS 9/5upn T(dsU (al[#S [%U 55 /nU HBunnnnT"UUS SS9nUR'UR)U5UR)U545 MD [+USS9upU R,(d1U R/S S9unn U R/S S9unn [1UU5nO [1S 5nU UR2-U UR4-- $![ a Us$f=fs sn f)a Simplifies a rational expression ``expr`` modulo the prime ideal generated by ``G``. ``G`` should be a Groebner basis of the ideal. Examples ======== >>> from sympy.simplify.ratsimp import ratsimpmodprime >>> from sympy.abc import x, y >>> eq = (x + y**5 + y)/(x - y) >>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex') (-x**2 - x*y - x - y)/(-x**2 + x*y) If ``polynomial`` is ``False``, the algorithm computes a rational simplification which minimizes the sum of the total degrees of the numerator and the denominator. If ``polynomial`` is ``True``, this function just brings numerator and denominator into a canonical form. This is much faster, but has potentially worse results. References ========== .. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial Ideal, https://dl.acm.org/doi/pdf/10.1145/1145768.1145809 (specifically, the second algorithm) r)solveratsimpmodprimez.Cannot compute rational simplification over %sNc>^US:XaS/$/n[[[TR55U5H_nS/[TR5-mUHnTU==S- ss'M [ U4SjT55(dMNUR T5 Ma UVs/sH%n[ U5R"TR6PM' snT"US- 5-$s snf)zs Compute all monomials with degree less than ``n`` that are not divisible by any element of ``leading_monomials``. rc3@># UHn[TU5SLv M g7fN)r).0lmgms r 5ratsimpmodprime..staircase..bs$&$58<3'4/$s)rrangelengensallappendr as_expr) nSmiisr*leading_monomialsopt staircases @rr:"ratsimpmodprime..staircaseVs 63J /c#((m0DaHBCM!A! &$&&& I9::1 ##SXX.:Yq1u=MMM:s,Cc0>^^^^XpeSnUR5UR5-nT(aUS- n OUn X4-U ::GaX44T;aGOTRX445 T"U5mT"U5m[SX4TT45 [S[ T5-[ S9m[S[ T5-[ S9mTT-n [ [UU4Sj[[ T5555TRU -5n [ [UU4Sj[[ T5555TRU -5n [X -X-- TTRU -TRS S 9Sn [ U TRS 9R5nT"UTT-S S S 9nU(Ga=[S UR555(GdU RU5nU RU5nUR[![#[%TT-S/[ T5[ T5--5555nUR[![#[%TT-S/[ T5[ T5--5555n[ UTR5n[ UTR5nUS:Xa ['S5eUR)XUTT-45 X4-U:waUS/nOUS- nUS- nUS- nX4-U ::aGMUS:aT"XVX#XG- 5upVnT"XVX#U- U5upVnXVU4$)a Computes a rational simplification of ``a/b`` which minimizes the sum of the total degrees of the numerator and the denominator. Explanation =========== The algorithm proceeds by looking at ``a * d - b * c`` modulo the ideal generated by ``G`` for some ``c`` and ``d`` with degree less than ``a`` and ``b`` respectively. The coefficients of ``c`` and ``d`` are indeterminates and thus the coefficients of the normalform of ``a * d - b * c`` are linear polynomials in these indeterminates. If these linear polynomials, considered as system of equations, have a nontrivial solution, then `\frac{a}{b} \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So, by construction, the degree of ``c`` and ``d`` is less than the degree of ``a`` and ``b``, so a simpler representation has been found. After a simpler representation has been found, the algorithm tries to reduce the degree of the numerator and denominator and returns the result afterwards. As an extension, if quick=False, we look at all possible degrees such that the total degree is less than *or equal to* the best current solution. We retain a list of all solutions of minimal degree, and try to find the best one at the end. rr%z%s / %s: %s, %szc:%d)clszd:%dc3:># UHnTUTU-v M g7fr')r(r6CsM1s rr+._ratsimpmodprime..:>aBqEBqEM>c3:># UHnTUTU-v M g7fr'r?)r(r6DsM2s rr+rBrCrDT)orderpolys)r/ particularrc3*# UH oS:Hv M g7f)rNr?)r(r7s rr+rBs<|!Av|szIdeal not prime?) total_degreeaddrrr.rr sumr-r/r rHcoeffsr0valuessubsdictlistzip ValueErrorr1)aballsolNDcdstepsmaxdegboundngc_hatd_hatrr4solr@rFrArGG_ratsimpmodprimer9rr!r:testeds @@@@rrg)ratsimpmodprime.._ratsimpmodprimehs:1!ANN$44 QJEEeunv JJv 1B1B $qRn 5#b')u5B#b')u5BbB:5R>::CHHrMKE:5R>::CHHrMKE AI-q#((R-!iit5568AQSXX&--/A27t4@C3!ratsimpmodprime..s'QqTZZ\):S1=N)Nrj)key)convert)rr)sympy.solvers.solversr!rrrr rrkhas_assoc_Field get_fieldrLMrHsetr r/r rr.r1rSminis_Field clear_denomsrqp)rrfrrr/argsnumdenomrIrkrr]r^rZnewsolrcrdr4rbrecndnrrgr8r9r!r:rhs `` @@@@@@rr"r"s9<, T",,.JC,c\A-=MMM sZZF %%'  rs23**'YY8B.!,+/5rj