LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUpZXF1YWxpdHlHRjQvJStleGVjdXRhYmxlR0Y0Ri8=with(LinearAlgebra):
with(Groebner):
# ---------------------------------------------------------------------------
# inv2ev
#
# Evaluate a polynomial P at the point defined by the Groebner basis B.
# 'mo' is the pre-built monomial order (passed in to avoid recomputation).
# ---------------------------------------------------------------------------
inv2ev := proc(P, B, mo)
NormalForm(P, B, mo);
end proc:
# ---------------------------------------------------------------------------
# create_groebner_basis
#
# Compute a Groebner basis for the ideal
# <J[i] - fI[i], minpolys[k], algfun_minpolys[l]>
# in the tdeg order 'mo'.
# Both 'minpolys' (for algebraic constants) and 'algfun_minpolys' (for
# algebraic functions) default to [] for backward compatibility.
# ---------------------------------------------------------------------------
create_groebner_basis := proc(J, fI, mo,
minpolys := [],
algfun_minpolys := [])
local i:
Basis(
[seq(J[i] - fI[i], i = 1..nops(J)),
op(minpolys),
op(algfun_minpolys)],
mo
);
end proc:
# ---------------------------------------------------------------------------
# invert_matrix_groebner
#
# Invert the matrix A modulo the ideal encoded by the Groebner basis B.
# 'mo' is the pre-built monomial order; 'n' is the matrix size.
# The scalar factor denomdet * Cdet[1][-1] is computed once outside
# the entry loop to avoid redundant NormalForm reductions.
# ---------------------------------------------------------------------------
invert_matrix_groebner := proc(A, B, n, mo)
local numerdet, denomdet, Bdet, Cdet, invA, scalar_factor, i, j:
numerdet := NormalForm(numer(Determinant(A)), B, mo):
denomdet := NormalForm(denom(Determinant(A)), B, mo):
Bdet, Cdet := Basis([op(B), numerdet], mo, output = extended):
scalar_factor := NormalForm(denomdet * Cdet[1][-1], B, mo):
invA := Adjoint(A):
for i to n do
for j to n do
invA[i,j] := NormalForm(scalar_factor * invA[i,j], B, mo)
od:
od:
invA;
end proc:
# ---------------------------------------------------------------------------
# compute_algfun_derivatives
#
# For each algebraic function variable g_l with minimal polynomial
# P_l(g_l) = 0 (P_l in Q(t)[x], passed as a polynomial in g_l),
# compute the total derivative g_l' = -P_d(g_l) / P'(g_l)
# in polynomial form by inverting P'(g_l) mod P_l(g_l) via gcdex.
#
# Arguments:
# algfuns -- list of algebraic function variables, e.g. [g1, g2]
# algfun_minpolys -- list of their minimal polynomials as polynomials in
# the corresponding variable with coefficients in Q(t),
# e.g. [g1^3 - t, g2^2 - t^2 + 1]
# t -- the derivation variable
# B -- Groebner basis (for final reduction)
# mo -- pre-built monomial order
#
# Returns a list [gd1, gd2, ...] where gd_l is the derivative of g_l,
# expressed as a polynomial in algfuns (degree < deg P_l) with coefficients
# in Q(t), reduced mod B.
# ---------------------------------------------------------------------------
compute_algfun_derivatives := proc(algfuns, algfun_minpolys, t, B, mo)
local result, l, g, P, Pd_poly, Pprime_poly, x, inv_Pprime, r, gcd_val:
result := []:
for l to nops(algfuns) do
g := algfuns[l]:
P := algfun_minpolys[l]: # polynomial in g with coeffs in Q(t)
# P_d: differentiate every coefficient of P w.r.t. t.
# Since P is a polynomial in g, diff(P, t) differentiates the
# coefficients and leaves g untouched (g is a Maple symbol here).
Pd_poly := diff(P, t):
# P': formal derivative of P with respect to g.
Pprime_poly := diff(P, g):
# Invert P'(g) modulo P(g) in Q(t)[g] via the extended gcdex.
# gcdex returns gcd and sets 's', 'r' such that
# s * Pprime_poly + r * P = gcd
# Since P is irreducible and deg(Pprime_poly) < deg(P), gcd = c in Q(t).
# Then s/c * Pprime_poly \342\211\241 1 (mod P).
gcd_val := gcdex(Pprime_poly, P, g, 'inv_Pprime', 'r'):
# g' = -inv_Pprime(g) * Pd(g) / gcd_val, reduced mod B.
result := [op(result),
NormalForm(
normal(-inv_Pprime * Pd_poly / gcd_val),
B, mo)]:
od:
result;
end proc:
# ---------------------------------------------------------------------------
# total_diff
#
# Compute the total derivative of 'expr' with respect to t, accounting for
# the fact that each algebraic function variable g_l depends on t with
# known derivative gd_l (precomputed by compute_algfun_derivatives).
#
# The result is reduced mod B with monomial order mo.
#
# When algfuns = [] this reduces to NormalForm(diff(expr, t), B, mo),
# recovering the previous behaviour exactly.
# ---------------------------------------------------------------------------
total_diff := proc(expr, t, algfuns, algfun_derivs, B, mo)
local d, l:
# Partial derivative w.r.t. t (treats all algebraic variables as constants)
d := diff(expr, t):
# Chain rule contributions from each algebraic function variable
for l to nops(algfuns) do
d := d + diff(expr, algfuns[l]) * algfun_derivs[l]:
od:
NormalForm(d, B, mo);
end proc:
# ---------------------------------------------------------------------------
# create_lode
#
# Given:
# J -- array of m generators of the G-invariant ring C[xs]^G
# fI -- array of m values for the generators; in full generality
# these are polynomials in Q(t)[algvars, algfuns]
# xs -- array of n solution-space variables
# DD -- differential operator domain, e.g. [Dt, t]
# algvars -- (optional) list of variables representing algebraic
# constants (algebraic over Q), e.g. [a]. Default: [].
# minpolys -- (optional) minimal polynomials for algvars,
# e.g. [a^2 - 2]. Default: [].
# algfuns -- (optional) list of variables representing algebraic
# functions (algebraic over Q(t)), e.g. [g]. Default: [].
# algfun_minpolys -- (optional) minimal polynomials for algfuns as
# polynomials in the corresponding variable with
# coefficients in Q(t), e.g. [g^2 - t^3 + 1].
# Default: [].
#
# All optional arguments default to [] so that every previous call site
# remains valid without modification.
#
# The three cases are:
# (1) algvars=[], algfuns=[] -- original algorithm, fI rational in t over Q
# (2) algvars<>[],algfuns=[] -- fI have algebraic constant coefficients
# (3) algfuns<>[] -- fI involve algebraic functions of t
#
# Returns the monic LODE of order n (as an operator in DD) whose Schwarz map
# parametrises the G-invariant curve defined by J[i] = fI[i].
# ---------------------------------------------------------------------------
create_lode := proc(J, fI, xs,
DD := _Envdiffopdomain,
algvars := [],
minpolys := [],
algfuns := [],
algfun_minpolys := [])
local n, t, Dt, allvars, mo, B, F, Fij, invFij, v,
algfun_derivs, xij, EQ_left, EQ_last, invEQ, c, cs, L,
i, j, k:
description
"Given generators J of the G-invariant ring and their values fI "
"(polynomials in Q(t)[algvars, algfuns], where algvars are algebraic "
"over Q with minimal polynomials minpolys, and algfuns are algebraic "
"over Q(t) with minimal polynomials algfun_minpolys), return the monic "
"LODE (as an operator in DD) whose Schwarz map parametrises the "
"G-invariant curve defined by J[i] = fI[i]. "
"All optional arguments default to [] for backward compatibility.":
# --- Input validation ---------------------------------------------------
if type(DD, symbol) then
error "differential algebra not specified"
fi:
if nops(algvars) <> nops(minpolys) then
error "algvars and minpolys must have the same length"
fi:
if nops(algfuns) <> nops(algfun_minpolys) then
error "algfuns and algfun_minpolys must have the same length"
fi:
n := nops(xs):
Dt := DD[1]:
t := DD[2]:
# --- Monomial order -----------------------------------------------------
# Variables: solution-space coords, then algebraic constants, then
# algebraic functions. Built ONCE and reused everywhere.
allvars := [op(xs), op(algvars), op(algfuns)]:
mo := tdeg(op(allvars)):
# --- Step 1: Groebner basis ---------------------------------------------
# Ideal: <J[i]-fI[i], minpolys[k], algfun_minpolys[l]> over Q.
B := create_groebner_basis(J, fI, mo, minpolys, algfun_minpolys):
# --- Step 2: Derivatives of algebraic function variables ----------------
# Precompute g_l' = -P_d(g_l) / P'(g_l) (mod P_l, mod B) for each g_l.
# When algfuns = [] this returns [] and costs nothing.
algfun_derivs := compute_algfun_derivatives(
algfuns, algfun_minpolys, t, B, mo):
# --- Step 3: Jacobian matrix of the first n invariants ------------------
F := [seq(J[i], i = 1..n)]:
Fij := Matrix(n):
for i to n do
for j to n do
Fij[i,j] := NormalForm(diff(F[i], xs[j]), B, mo)
od
od:
# --- Step 4: First-order system -----------------------------------------
# xij[i,1] = (J^{-1} * D_t(fI))[i], reduced mod B.
# D_t(fI[i]) is the TOTAL derivative w.r.t. t, accounting for any g_l(t).
invFij := invert_matrix_groebner(Fij, B, n, mo):
v := invFij . Vector(
[seq(total_diff(fI[i], t, algfuns, algfun_derivs, B, mo),
i = 1..n)]):
xij := Matrix(n):
for i to n do
xij[i,1] := NormalForm(v[i], B, mo)
od:
# --- Step 5: Differentiation loop ---------------------------------------
# xij[i,j] = d^{j-1}/dt^{j-1} xs[i], built recursively via the chain
# rule and reduced mod B at each step.
# total_diff handles the g_l(t) contributions; diff(..., xs[k]) gives
# the xs-gradient as before.
# The total derivative of xij[i,j-1] w.r.t. t splits into:
# (a) xs-gradient: sum_k (d/d xs[k]) xij[i,j-1] * xij[k,1]
# (b) explicit t-dependence + chain-rule through any g_l(t):
# total_diff(xij[i,j-1], t, algfuns, algfun_derivs, B, mo)
# Both pieces are reduced mod B at each step.
for j from 2 to n do
for i to n do
xij[i,j] := NormalForm(
add(diff(xij[i,j-1], xs[k]) * xij[k,1], k = 1..n)
+ total_diff(xij[i,j-1], t, algfuns, algfun_derivs, B, mo),
B, mo):
od
od:
# --- Step 6: LODE coefficients ------------------------------------------
# Invert [xs | xij[*,1..n-1]] to express xij[*,n] as a linear combination.
EQ_left := Matrix(n, n, (i,j) -> `if`(j = 1, xs[i], xij[i,j-1])):
EQ_last := Vector(n, i -> xij[i,n]):
invEQ := invert_matrix_groebner(EQ_left, B, n, mo):
c := invEQ . EQ_last:
# --- Step 7: Assemble the operator --------------------------------------
# Evaluate each coefficient at the curve point (reduce mod B) and
# factor. After reduction mod B: algvars have degree < deg(minpoly),
# algfuns have degree < deg(algfun_minpoly) -- standard representations
# in Q(alpha, g).
cs := [-seq(inv2ev(c[k], B, mo), k = 1..n)]:
L := add(factor(cs[i]) * Dt^(i-1), i = 1..n) + Dt^n;
end proc:with(NumberTheory):
P:=CyclotomicPolynomial(18,z);
alias(zeta=RootOf(P=0,index=1));f[4] := x[0]^3*x[1]+(-z^3+1)*x[1]*x[2]^3+(-3*z^5)*x[0]*x[1]*x[2]*x[3]+x[1]*x[3]^3:
f[6] := 3*x[1]^6+(5*z^4-5*z)*x[0]^4*x[2]^2+(2*z^4)*x[0]*x[2]^5+(-2*z^3+2)*x[0]^5*x[3]+(-20*z^3)*x[0]^2*x[2]^3*x[3]+(20*z^2)*x[0]^3*x[2]*x[3]^2+(-5*z^5+5*z^2)*x[2]^4*x[3]^2+(20*z^4-20*z)*x[0]*x[2]^2*x[3]^3+(-5*z^3+5)*x[0]^2*x[3]^4+(2*z^2)*x[2]*x[3]^5:
f[12] := 81*x[1]^12+(-4*z^5+4*z^2)*x[0]^11*x[2]+(-165*z^5)*x[0]^8*x[2]^4+(-264*z^2)*x[0]^5*x[2]^7+(22*z^5-22*z^2)*x[0]^2*x[2]^10+(220*z^4)*x[0]^9*x[2]^2*x[3]+(1848*z)*x[0]^6*x[2]^5*x[3]+(-660*z^4+660*z)*x[0]^3*x[2]^8*x[3]+(-4*z^4)*x[2]^11*x[3]+(-22*z^3)*x[0]^10*x[3]^2-2640*x[0]^7*x[2]^3*x[3]^2+(4620*z^3-4620)*x[0]^4*x[2]^6*x[3]^2+(220*z^3)*x[0]*x[2]^9*x[3]^2+(-660*z^5+660*z^2)*x[0]^8*x[2]*x[3]^3+(-9240*z^5)*x[0]^5*x[2]^4*x[3]^3+(-2640*z^2)*x[0]^2*x[2]^7*x[3]^3+(4620*z^4)*x[0]^6*x[2]^2*x[3]^4+(9240*z)*x[0]^3*x[2]^5*x[3]^4+(-165*z^4+165*z)*x[2]^8*x[3]^4+(-264*z^3)*x[0]^7*x[3]^5-9240*x[0]^4*x[2]^3*x[3]^5+(1848*z^3-1848)*x[0]*x[2]^6*x[3]^5+(-1848*z^5+1848*z^2)*x[0]^5*x[2]*x[3]^6+(-4620*z^5)*x[0]^2*x[2]^4*x[3]^6+(2640*z^4)*x[0]^3*x[2]^2*x[3]^7+(264*z)*x[2]^5*x[3]^7+(-165*z^3)*x[0]^4*x[3]^8-660*x[0]*x[2]^3*x[3]^8+(-220*z^5+220*z^2)*x[0]^2*x[2]*x[3]^9+(22*z^4)*x[2]^2*x[3]^10+(-4*z^3)*x[0]*x[3]^11:
f[18] := 528821*x[0]^18+36756720*x[0]^9*x[1]^9-729*x[1]^18+(-164402784*z^5+164402784*z^2)*x[0]^11*x[1]^6*x[2]+(594864*z^5-594864*z^2)*x[0]^2*x[1]^15*x[2]+(-60918480*z^4+60918480*z)*x[0]^13*x[1]^3*x[2]^2+(67665780*z^4-67665780*z)*x[0]^4*x[1]^12*x[2]^2+(-425136*z^3+425136)*x[0]^15*x[2]^3+(882161280*z^3-882161280)*x[0]^6*x[1]^9*x[2]^3+(1902160260*z^5)*x[0]^8*x[1]^6*x[2]^4+(485188704*z^4)*x[0]^10*x[1]^3*x[2]^5+(-27066312*z^4)*x[0]*x[1]^12*x[2]^5+(6534528*z^3)*x[0]^12*x[2]^6+(-110270160*z^3)*x[0]^3*x[1]^9*x[2]^6+(-926269344*z^2)*x[0]^5*x[1]^6*x[2]^7+(-283551840*z)*x[0]^7*x[1]^3*x[2]^8+5299580*x[0]^9*x[2]^9-36756720*x[1]^9*x[2]^9+(-187459272*z^5+187459272*z^2)*x[0]^2*x[1]^6*x[2]^10+(10024560*z^4-10024560*z)*x[0]^4*x[1]^3*x[2]^11+(-3360084*z^3+3360084)*x[0]^6*x[2]^12+(-7050240*z^4)*x[0]*x[1]^3*x[2]^14+(-44880*z^3)*x[0]^3*x[2]^15+528821*x[2]^18+(-7050240*z^3+7050240)*x[0]^14*x[1]^3*x[3]+(27066312*z^3-27066312)*x[0]^5*x[1]^12*x[3]+(-145350*z^5)*x[0]^16*x[2]*x[3]+(661620960*z^5)*x[0]^7*x[1]^9*x[2]*x[3]+(2095133040*z^4)*x[0]^9*x[1]^6*x[2]^2*x[3]+(-594864*z^4)*x[1]^15*x[2]^2*x[3]+(721768320*z^3)*x[0]^11*x[1]^3*x[2]^3*x[3]+(10110240*z^2)*x[0]^13*x[2]^4*x[3]+(-1654052400*z^2)*x[0]^4*x[1]^9*x[2]^4*x[3]+(-4631346720*z)*x[0]^6*x[1]^6*x[2]^5*x[3]-1654052400*x[0]^8*x[1]^3*x[2]^6*x[3]+(-13302432*z^5+13302432*z^2)*x[0]^10*x[2]^7*x[3]+(-661620960*z^5+661620960*z^2)*x[0]*x[1]^9*x[2]^7*x[3]+(661620960*z^4-661620960*z)*x[0]^3*x[1]^6*x[2]^8*x[3]+(-441080640*z^3+441080640)*x[0]^5*x[1]^3*x[2]^9*x[3]+(-32205888*z^5)*x[0]^7*x[2]^10*x[3]+(-164402784*z^4)*x[1]^6*x[2]^11*x[3]+(150368400*z^3)*x[0]^2*x[1]^3*x[2]^12*x[3]+(-4326840*z^2)*x[0]^4*x[2]^13*x[3]+(-145350*z^5+145350*z^2)*x[0]*x[2]^16*x[3]+(187459272*z^3)*x[0]^10*x[1]^6*x[3]^2+(594864*z^3)*x[0]*x[1]^15*x[3]^2+(150368400*z^2)*x[0]^12*x[1]^3*x[2]*x[3]^2+(844560*z)*x[0]^14*x[2]^2*x[3]^2+(-992431440*z)*x[0]^5*x[1]^9*x[2]^2*x[3]^2-5292967680*x[0]^7*x[1]^6*x[2]^3*x[3]^2+(2205403200*z^5-2205403200*z^2)*x[0]^9*x[1]^3*x[2]^4*x[3]^2+(-67665780*z^5+67665780*z^2)*x[1]^12*x[2]^4*x[3]^2+(77523264*z^4-77523264*z)*x[0]^11*x[2]^5*x[3]^2+(992431440*z^4-992431440*z)*x[0]^2*x[1]^9*x[2]^5*x[3]^2+(-4631346720*z^3+4631346720)*x[0]^4*x[1]^6*x[2]^6*x[3]^2+(-1323241920*z^5)*x[0]^6*x[1]^3*x[2]^7*x[3]^2+(139806810*z^4)*x[0]^8*x[2]^8*x[3]^2+(2095133040*z^3)*x[0]*x[1]^6*x[2]^9*x[3]^2+(-661620960*z^2)*x[0]^3*x[1]^3*x[2]^10*x[3]^2+(59479056*z)*x[0]^5*x[2]^11*x[3]^2+(-60918480*z^5+60918480*z^2)*x[1]^3*x[2]^13*x[3]^2+(844560*z^4-844560*z)*x[0]^2*x[2]^14*x[3]^2-44880*x[0]^15*x[3]^3+110270160*x[0]^6*x[1]^9*x[3]^3+(661620960*z^5-661620960*z^2)*x[0]^8*x[1]^6*x[2]*x[3]^3+(661620960*z^4-661620960*z)*x[0]^10*x[1]^3*x[2]^2*x[3]^3+(70543200*z^3-70543200)*x[0]^12*x[2]^3*x[3]^3+(-2205403200*z^3+2205403200)*x[0]^3*x[1]^9*x[2]^3*x[3]^3+(-4631346720*z^5)*x[0]^5*x[1]^6*x[2]^4*x[3]^3+(-2646483840*z^4)*x[0]^7*x[1]^3*x[2]^5*x[3]^3+(628948320*z^3)*x[0]^9*x[2]^6*x[3]^3+(882161280*z^3)*x[1]^9*x[2]^6*x[3]^3+(-5292967680*z^2)*x[0]^2*x[1]^6*x[2]^7*x[3]^3+(1654052400*z)*x[0]^4*x[1]^3*x[2]^8*x[3]^3+665705040*x[0]^6*x[2]^9*x[3]^3+(721768320*z^4-721768320*z)*x[0]*x[1]^3*x[2]^11*x[3]^3+(-70543200*z^3+70543200)*x[0]^3*x[2]^12*x[3]^3+(425136*z^3)*x[2]^15*x[3]^3+(10024560*z^3-10024560)*x[0]^11*x[1]^3*x[3]^4+(-67665780*z^3+67665780)*x[0]^2*x[1]^12*x[3]^4+(4326840*z^5)*x[0]^13*x[2]*x[3]^4+(-1654052400*z^5)*x[0]^4*x[1]^9*x[2]*x[3]^4+(-4631346720*z^4)*x[0]^6*x[1]^6*x[2]^2*x[3]^4+(-1654052400*z^3)*x[0]^8*x[1]^3*x[2]^3*x[3]^4+(226666440*z^2)*x[0]^10*x[2]^4*x[3]^4+(-1654052400*z^2)*x[0]*x[1]^9*x[2]^4*x[3]^4+(4631346720*z)*x[0]^3*x[1]^6*x[2]^5*x[3]^4+(-871659360*z^5+871659360*z^2)*x[0]^7*x[2]^7*x[3]^4+(1902160260*z^4-1902160260*z)*x[1]^6*x[2]^8*x[3]^4+(-2205403200*z^3+2205403200)*x[0]^2*x[1]^3*x[2]^9*x[3]^4+(-226666440*z^5)*x[0]^4*x[2]^10*x[3]^4+(-10110240*z^2)*x[0]*x[2]^13*x[3]^4+(-926269344*z^3)*x[0]^7*x[1]^6*x[3]^5+(-441080640*z^2)*x[0]^9*x[1]^3*x[2]*x[3]^5+(-27066312*z^2)*x[1]^12*x[2]*x[3]^5+(-59479056*z)*x[0]^11*x[2]^2*x[3]^5+(-992431440*z)*x[0]^2*x[1]^9*x[2]^2*x[3]^5-4631346720*x[0]^4*x[1]^6*x[2]^3*x[3]^5+(882161280*z^4-882161280*z)*x[0]^8*x[2]^5*x[3]^5+(-4631346720*z^3+4631346720)*x[0]*x[1]^6*x[2]^6*x[3]^5+(2646483840*z^5)*x[0]^3*x[1]^3*x[2]^7*x[3]^5+(882161280*z^4)*x[0]^5*x[2]^8*x[3]^5+(-485188704*z^2)*x[1]^3*x[2]^10*x[3]^5+(77523264*z)*x[0]^2*x[2]^11*x[3]^5-3360084*x[0]^12*x[3]^6-882161280*x[0]^3*x[1]^9*x[3]^6+(4631346720*z^5-4631346720*z^2)*x[0]^5*x[1]^6*x[2]*x[3]^6+(1323241920*z^4-1323241920*z)*x[0]^7*x[1]^3*x[2]^2*x[3]^6+(665705040*z^3-665705040)*x[0]^9*x[2]^3*x[3]^6+(-110270160*z^3+110270160)*x[1]^9*x[2]^3*x[3]^6+(4631346720*z^5)*x[0]^2*x[1]^6*x[2]^4*x[3]^6+(2795961168*z^3)*x[0]^6*x[2]^6*x[3]^6+(1654052400*z)*x[0]*x[1]^3*x[2]^8*x[3]^6+628948320*x[0]^3*x[2]^9*x[3]^6+(-6534528*z^3+6534528)*x[2]^12*x[3]^6+(283551840*z^3-283551840)*x[0]^8*x[1]^3*x[3]^7+(32205888*z^5)*x[0]^10*x[2]*x[3]^7+(661620960*z^5)*x[0]*x[1]^9*x[2]*x[3]^7+(5292967680*z^4)*x[0]^3*x[1]^6*x[2]^2*x[3]^7+(2646483840*z^3)*x[0]^5*x[1]^3*x[2]^3*x[3]^7+(871659360*z^2)*x[0]^7*x[2]^4*x[3]^7+(926269344*z)*x[1]^6*x[2]^5*x[3]^7-1323241920*x[0]^2*x[1]^3*x[2]^6*x[3]^7+(-871659360*z^5+871659360*z^2)*x[0]^4*x[2]^7*x[3]^7+(-13302432*z^5)*x[0]*x[2]^10*x[3]^7+(1902160260*z^3)*x[0]^4*x[1]^6*x[3]^8+(1654052400*z^2)*x[0]^6*x[1]^3*x[2]*x[3]^8+(-139806810*z)*x[0]^8*x[2]^2*x[3]^8+661620960*x[0]*x[1]^6*x[2]^3*x[3]^8+(-1654052400*z^5+1654052400*z^2)*x[0]^3*x[1]^3*x[2]^4*x[3]^8+(882161280*z^4-882161280*z)*x[0]^5*x[2]^5*x[3]^8+(283551840*z^5)*x[1]^3*x[2]^7*x[3]^8+(139806810*z^4)*x[0]^2*x[2]^8*x[3]^8-5299580*x[0]^9*x[3]^9+36756720*x[1]^9*x[3]^9+(-2095133040*z^5+2095133040*z^2)*x[0]^2*x[1]^6*x[2]*x[3]^9+(-2205403200*z^4+2205403200*z)*x[0]^4*x[1]^3*x[2]^2*x[3]^9+(628948320*z^3-628948320)*x[0]^6*x[2]^3*x[3]^9+(-441080640*z^4)*x[0]*x[1]^3*x[2]^5*x[3]^9+(665705040*z^3)*x[0]^3*x[2]^6*x[3]^9+5299580*x[2]^9*x[3]^9+(-485188704*z^3+485188704)*x[0]^5*x[1]^3*x[3]^10+(13302432*z^5)*x[0]^7*x[2]*x[3]^10+(-187459272*z^4)*x[1]^6*x[2]^2*x[3]^10+(-661620960*z^3)*x[0]^2*x[1]^3*x[2]^3*x[3]^10+(226666440*z^2)*x[0]^4*x[2]^4*x[3]^10+(-32205888*z^5+32205888*z^2)*x[0]*x[2]^7*x[3]^10+(-164402784*z^3)*x[0]*x[1]^6*x[3]^11+(-721768320*z^2)*x[0]^3*x[1]^3*x[2]*x[3]^11+(-77523264*z)*x[0]^5*x[2]^2*x[3]^11+(10024560*z^5-10024560*z^2)*x[1]^3*x[2]^4*x[3]^11+(59479056*z^4-59479056*z)*x[0]^2*x[2]^5*x[3]^11-6534528*x[0]^6*x[3]^12+(150368400*z^4-150368400*z)*x[0]*x[1]^3*x[2]^2*x[3]^12+(70543200*z^3-70543200)*x[0]^3*x[2]^3*x[3]^12+(3360084*z^3)*x[2]^6*x[3]^12+(60918480*z^3-60918480)*x[0]^2*x[1]^3*x[3]^13+(10110240*z^5)*x[0]^4*x[2]*x[3]^13+(4326840*z^2)*x[0]*x[2]^4*x[3]^13+(7050240*z^2)*x[1]^3*x[2]*x[3]^14+(844560*z)*x[0]^2*x[2]^2*x[3]^14+425136*x[0]^3*x[3]^15+(44880*z^3-44880)*x[2]^3*x[3]^15+(-145350*z^5)*x[0]*x[2]*x[3]^16+528821*x[3]^18:
s[12] := x[0]^9*x[1]^3+27/1595*x[1]^12+(-177/319*z^5+177/319*z^2)*x[0]^11*x[2]+(-81/29*z^5+81/29*z^2)*x[0]^2*x[1]^9*x[2]+(-756/29*z^4+756/29*z)*x[0]^4*x[1]^6*x[2]^2+(-168/29*z^3+168/29)*x[0]^6*x[1]^3*x[2]^3+(18/29*z^5)*x[0]^8*x[2]^4+(756/145*z^4)*x[0]*x[1]^6*x[2]^5+(588/29*z^3)*x[0]^3*x[1]^3*x[2]^6+(-36/29*z^2)*x[0]^5*x[2]^7-x[1]^3*x[2]^9+(-3/145*z^5+3/145*z^2)*x[0]^2*x[2]^10+(-756/145*z^3+756/145)*x[0]^5*x[1]^6*x[3]+(576/29*z^5)*x[0]^7*x[1]^3*x[2]*x[3]+(21/29*z^4)*x[0]^9*x[2]^2*x[3]+(81/29*z^4)*x[1]^9*x[2]^2*x[3]+(756/29*z^3)*x[0]^2*x[1]^6*x[2]^3*x[3]+(-630/29*z^2)*x[0]^4*x[1]^3*x[2]^4*x[3]+(-1386/145*z)*x[0]^6*x[2]^5*x[3]+(-576/29*z^5+576/29*z^2)*x[0]*x[1]^3*x[2]^7*x[3]+(126/29*z^4-126/29*z)*x[0]^3*x[2]^8*x[3]+(-177/319*z^4)*x[2]^11*x[3]+(3/145*z^3)*x[0]^10*x[3]^2+(-81/29*z^3)*x[0]*x[1]^9*x[3]^2+(-756/29*z^2)*x[0]^3*x[1]^6*x[2]*x[3]^2+(-1512/29*z)*x[0]^5*x[1]^3*x[2]^2*x[3]^2-468/29*x[0]^7*x[2]^3*x[3]^2+(756/29*z^5-756/29*z^2)*x[1]^6*x[2]^4*x[3]^2+(1512/29*z^4-1512/29*z)*x[0]^2*x[1]^3*x[2]^5*x[3]^2+(-126/29*z^3+126/29)*x[0]^4*x[2]^6*x[3]^2+(21/29*z^3)*x[0]*x[2]^9*x[3]^2-588/29*x[0]^6*x[1]^3*x[3]^3+(126/29*z^5-126/29*z^2)*x[0]^8*x[2]*x[3]^3+(-756/29*z^4+756/29*z)*x[0]*x[1]^6*x[2]^2*x[3]^3+(-3360/29*z^3+3360/29)*x[0]^3*x[1]^3*x[2]^3*x[3]^3+(-126/29*z^5)*x[0]^5*x[2]^4*x[3]^3+(-168/29*z^3)*x[1]^3*x[2]^6*x[3]^3+(-468/29*z^2)*x[0]^2*x[2]^7*x[3]^3+(756/29*z^3-756/29)*x[0]^2*x[1]^6*x[3]^4+(-630/29*z^5)*x[0]^4*x[1]^3*x[2]*x[3]^4+(-126/29*z^4)*x[0]^6*x[2]^2*x[3]^4+(-630/29*z^2)*x[0]*x[1]^3*x[2]^4*x[3]^4+(126/29*z)*x[0]^3*x[2]^5*x[3]^4+(18/29*z^4-18/29*z)*x[2]^8*x[3]^4+(-36/29*z^3)*x[0]^7*x[3]^5+(756/145*z^2)*x[1]^6*x[2]*x[3]^5+(-1512/29*z)*x[0]^2*x[1]^3*x[2]^2*x[3]^5-126/29*x[0]^4*x[2]^3*x[3]^5+(-1386/145*z^3+1386/145)*x[0]*x[2]^6*x[3]^5+168/29*x[0]^3*x[1]^3*x[3]^6+(1386/145*z^5-1386/145*z^2)*x[0]^5*x[2]*x[3]^6+(588/29*z^3-588/29)*x[1]^3*x[2]^3*x[3]^6+(126/29*z^5)*x[0]^2*x[2]^4*x[3]^6+(576/29*z^5)*x[0]*x[1]^3*x[2]*x[3]^7+(468/29*z^4)*x[0]^3*x[2]^2*x[3]^7+(36/29*z)*x[2]^5*x[3]^7+(18/29*z^3)*x[0]^4*x[3]^8+126/29*x[0]*x[2]^3*x[3]^8+x[1]^3*x[3]^9+(-21/29*z^5+21/29*z^2)*x[0]^2*x[2]*x[3]^9+(-3/145*z^4)*x[2]^2*x[3]^10+(-177/319*z^3)*x[0]*x[3]^11:
t[12] := x[0]^9*x[1]^3-27/205*x[1]^12+(21/205*z^5-21/205*z^2)*x[0]^11*x[2]+(1539/205*z^5-1539/205*z^2)*x[0]^2*x[1]^9*x[2]+(-1512/205*z^4+1512/205*z)*x[0]^4*x[1]^6*x[2]^2+(924/205*z^3-924/205)*x[0]^6*x[1]^3*x[2]^3+(-558/205*z^5)*x[0]^8*x[2]^4+(3024/205*z^4)*x[0]*x[1]^6*x[2]^5+(6972/205*z^3)*x[0]^3*x[1]^3*x[2]^6+(-144/41*z^2)*x[0]^5*x[2]^7-x[1]^3*x[2]^9+(141/205*z^5-141/205*z^2)*x[0]^2*x[2]^10+(-3024/205*z^3+3024/205)*x[0]^5*x[1]^6*x[3]+(2664/205*z^5)*x[0]^7*x[1]^3*x[2]*x[3]+(-2127/205*z^4)*x[0]^9*x[2]^2*x[3]+(-1539/205*z^4)*x[1]^9*x[2]^2*x[3]+(-12096/205*z^3)*x[0]^2*x[1]^6*x[2]^3*x[3]+(-4410/41*z^2)*x[0]^4*x[1]^3*x[2]^4*x[3]+(882/205*z)*x[0]^6*x[2]^5*x[3]+(-2664/205*z^5+2664/205*z^2)*x[0]*x[1]^3*x[2]^7*x[3]+(-234/205*z^4+234/205*z)*x[0]^3*x[2]^8*x[3]+(21/205*z^4)*x[2]^11*x[3]+(-141/205*z^3)*x[0]^10*x[3]^2+(1539/205*z^3)*x[0]*x[1]^9*x[3]^2+(12096/205*z^2)*x[0]^3*x[1]^6*x[2]*x[3]^2+(8316/205*z)*x[0]^5*x[1]^3*x[2]^2*x[3]^2+3168/205*x[0]^7*x[2]^3*x[3]^2+(1512/205*z^5-1512/205*z^2)*x[1]^6*x[2]^4*x[3]^2+(-8316/205*z^4+8316/205*z)*x[0]^2*x[1]^3*x[2]^5*x[3]^2+(630/41*z^3-630/41)*x[0]^4*x[2]^6*x[3]^2+(-2127/205*z^3)*x[0]*x[2]^9*x[3]^2-6972/205*x[0]^6*x[1]^3*x[3]^3+(-234/205*z^5+234/205*z^2)*x[0]^8*x[2]*x[3]^3+(12096/205*z^4-12096/205*z)*x[0]*x[1]^6*x[2]^2*x[3]^3+(3696/41*z^3-3696/41)*x[0]^3*x[1]^3*x[2]^3*x[3]^3+(1386/41*z^5)*x[0]^5*x[2]^4*x[3]^3+(924/205*z^3)*x[1]^3*x[2]^6*x[3]^3+(3168/205*z^2)*x[0]^2*x[2]^7*x[3]^3+(1512/205*z^3-1512/205)*x[0]^2*x[1]^6*x[3]^4+(-4410/41*z^5)*x[0]^4*x[1]^3*x[2]*x[3]^4+(630/41*z^4)*x[0]^6*x[2]^2*x[3]^4+(-4410/41*z^2)*x[0]*x[1]^3*x[2]^4*x[3]^4+(-1386/41*z)*x[0]^3*x[2]^5*x[3]^4+(-558/205*z^4+558/205*z)*x[2]^8*x[3]^4+(-144/41*z^3)*x[0]^7*x[3]^5+(3024/205*z^2)*x[1]^6*x[2]*x[3]^5+(8316/205*z)*x[0]^2*x[1]^3*x[2]^2*x[3]^5+1386/41*x[0]^4*x[2]^3*x[3]^5+(882/205*z^3-882/205)*x[0]*x[2]^6*x[3]^5-924/205*x[0]^3*x[1]^3*x[3]^6+(-882/205*z^5+882/205*z^2)*x[0]^5*x[2]*x[3]^6+(6972/205*z^3-6972/205)*x[1]^3*x[2]^3*x[3]^6+(-630/41*z^5)*x[0]^2*x[2]^4*x[3]^6+(2664/205*z^5)*x[0]*x[1]^3*x[2]*x[3]^7+(-3168/205*z^4)*x[0]^3*x[2]^2*x[3]^7+(144/41*z)*x[2]^5*x[3]^7+(-558/205*z^3)*x[0]^4*x[3]^8-234/205*x[0]*x[2]^3*x[3]^8+x[1]^3*x[3]^9+(2127/205*z^5-2127/205*z^2)*x[0]^2*x[2]*x[3]^9+(141/205*z^4)*x[2]^2*x[3]^10+(21/205*z^3)*x[0]*x[3]^11:
s[18] := x[0]^18-216216/95*x[0]^9*x[1]^9+729/475*x[1]^18+(-4186728/475*z^5+4186728/475*z^2)*x[0]^11*x[1]^6*x[2]+(-17496/25*z^5+17496/25*z^2)*x[0]^2*x[1]^15*x[2]+(636552/95*z^4-636552/95*z)*x[0]^13*x[1]^3*x[2]^2+(-2918916/95*z^4+2918916/95*z)*x[0]^4*x[1]^12*x[2]^2+(36312/95*z^3-36312/95)*x[0]^15*x[2]^3+(-5189184/95*z^3+5189184/95)*x[0]^6*x[1]^9*x[2]^3+(16540524/95*z^5)*x[0]^8*x[1]^6*x[2]^4+(-9513504/475*z^4)*x[0]^10*x[1]^3*x[2]^5+(1061424/95*z^4)*x[0]*x[1]^12*x[2]^5+(-297024/475*z^3)*x[0]^12*x[2]^6+(648648/95*z^3)*x[0]^3*x[1]^9*x[2]^6+(-159567408/475*z^2)*x[0]^5*x[1]^6*x[2]^7+(-926640/19*z)*x[0]^7*x[1]^3*x[2]^8+199628/95*x[0]^9*x[2]^9+216216/95*x[1]^9*x[2]^9+(-26594568/475*z^5+26594568/475*z^2)*x[0]^2*x[1]^6*x[2]^10+(570024/95*z^4-570024/95*z)*x[0]^4*x[1]^3*x[2]^11+(-1092*z^3+1092)*x[0]^6*x[2]^12+(1728/95*z^4)*x[0]*x[1]^3*x[2]^14+(-7368/95*z^3)*x[0]^3*x[2]^15+x[2]^18+(1728/95*z^3-1728/95)*x[0]^14*x[1]^3*x[3]+(-1061424/95*z^3+1061424/95)*x[0]^5*x[1]^12*x[3]+(-509562/475*z^5)*x[0]^16*x[2]*x[3]+(-3891888/95*z^5)*x[0]^7*x[1]^9*x[2]*x[3]+(5189184/95*z^4)*x[0]^9*x[1]^6*x[2]^2*x[3]+(17496/25*z^4)*x[1]^15*x[2]^2*x[3]+(-2044224/19*z^3)*x[0]^11*x[1]^3*x[2]^3*x[3]+(530712/95*z^3)*x[0]^2*x[1]^12*x[2]^3*x[3]+(-600768/95*z^2)*x[0]^13*x[2]^4*x[3]+(1945944/19*z^2)*x[0]^4*x[1]^9*x[2]^4*x[3]+(54486432/475*z)*x[0]^6*x[1]^6*x[2]^5*x[3]+18810792/95*x[0]^8*x[1]^3*x[2]^6*x[3]+(267696/475*z^5-267696/475*z^2)*x[0]^10*x[2]^7*x[3]+(3891888/95*z^5-3891888/95*z^2)*x[0]*x[1]^9*x[2]^7*x[3]+(25297272/95*z^4-25297272/95*z)*x[0]^3*x[1]^6*x[2]^8*x[3]+(-11243232/95*z^3+11243232/95)*x[0]^5*x[1]^3*x[2]^9*x[3]+(998712/95*z^5)*x[0]^7*x[2]^10*x[3]+(-4186728/475*z^4)*x[1]^6*x[2]^11*x[3]+(609336/95*z^3)*x[0]^2*x[1]^3*x[2]^12*x[3]+(219744/95*z^2)*x[0]^4*x[2]^13*x[3]+(-509562/475*z^5+509562/475*z^2)*x[0]*x[2]^16*x[3]+(26594568/475*z^3)*x[0]^10*x[1]^6*x[3]^2+(-17496/25*z^3)*x[0]*x[1]^15*x[3]^2+(609336/95*z^2)*x[0]^12*x[1]^3*x[2]*x[3]^2+(-530712/95*z^2)*x[0]^3*x[1]^12*x[2]*x[3]^2+(-346464/95*z)*x[0]^14*x[2]^2*x[3]^2+(5837832/95*z)*x[0]^5*x[1]^9*x[2]^2*x[3]^2-15567552/95*x[0]^7*x[1]^6*x[2]^3*x[3]^2+(-7351344/19*z^5+7351344/19*z^2)*x[0]^9*x[1]^3*x[2]^4*x[3]^2+(2918916/95*z^5-2918916/95*z^2)*x[1]^12*x[2]^4*x[3]^2+(8923824/475*z^4-8923824/475*z)*x[0]^11*x[2]^5*x[3]^2+(-5837832/95*z^4+5837832/95*z)*x[0]^2*x[1]^9*x[2]^5*x[3]^2+(-54486432/95*z^3+54486432/95)*x[0]^4*x[1]^6*x[2]^6*x[3]^2+(44108064/95*z^5)*x[0]^6*x[1]^3*x[2]^7*x[3]^2+(254826/5*z^4)*x[0]^8*x[2]^8*x[3]^2+(5189184/95*z^3)*x[0]*x[1]^6*x[2]^9*x[3]^2+(-5621616/95*z^2)*x[0]^3*x[1]^3*x[2]^10*x[3]^2+(3164616/95*z)*x[0]^5*x[2]^11*x[3]^2+(636552/95*z^5-636552/95*z^2)*x[1]^3*x[2]^13*x[3]^2+(-346464/95*z^4+346464/95*z)*x[0]^2*x[2]^14*x[3]^2-7368/95*x[0]^15*x[3]^3-648648/95*x[0]^6*x[1]^9*x[3]^3+(25297272/95*z^5-25297272/95*z^2)*x[0]^8*x[1]^6*x[2]*x[3]^3+(5621616/95*z^4-5621616/95*z)*x[0]^10*x[1]^3*x[2]^2*x[3]^3+(-530712/95*z^4+530712/95*z)*x[0]*x[1]^12*x[2]^2*x[3]^3+(816816/95*z^3-816816/95)*x[0]^12*x[2]^3*x[3]^3+(2594592/19*z^3-2594592/19)*x[0]^3*x[1]^9*x[2]^3*x[3]^3+(27243216/95*z^5)*x[0]^5*x[1]^6*x[2]^4*x[3]^3+(36324288/95*z^4)*x[0]^7*x[1]^3*x[2]^5*x[3]^3+(4012008/95*z^3)*x[0]^9*x[2]^6*x[3]^3+(-5189184/95*z^3)*x[1]^9*x[2]^6*x[3]^3+(-15567552/95*z^2)*x[0]^2*x[1]^6*x[2]^7*x[3]^3+(648648/19*z)*x[0]^4*x[1]^3*x[2]^8*x[3]^3+240240/19*x[0]^6*x[2]^9*x[3]^3+(-2044224/19*z^4+2044224/19*z)*x[0]*x[1]^3*x[2]^11*x[3]^3+(-816816/95*z^3+816816/95)*x[0]^3*x[2]^12*x[3]^3+(-36312/95*z^3)*x[2]^15*x[3]^3+(570024/95*z^3-570024/95)*x[0]^11*x[1]^3*x[3]^4+(2918916/95*z^3-2918916/95)*x[0]^2*x[1]^12*x[3]^4+(-219744/95*z^5)*x[0]^13*x[2]*x[3]^4+(1945944/19*z^5)*x[0]^4*x[1]^9*x[2]*x[3]^4+(-54486432/95*z^4)*x[0]^6*x[1]^6*x[2]^2*x[3]^4+(-648648/19*z^3)*x[0]^8*x[1]^3*x[2]^3*x[3]^4+(-2090088/95*z^2)*x[0]^10*x[2]^4*x[3]^4+(1945944/19*z^2)*x[0]*x[1]^9*x[2]^4*x[3]^4+(-27243216/95*z)*x[0]^3*x[1]^6*x[2]^5*x[3]^4-36324288/95*x[0]^5*x[1]^3*x[2]^6*x[3]^4+(-3953664/95*z^5+3953664/95*z^2)*x[0]^7*x[2]^7*x[3]^4+(16540524/95*z^4-16540524/95*z)*x[1]^6*x[2]^8*x[3]^4+(7351344/19*z^3-7351344/19)*x[0]^2*x[1]^3*x[2]^9*x[3]^4+(2090088/95*z^5)*x[0]^4*x[2]^10*x[3]^4+(600768/95*z^2)*x[0]*x[2]^13*x[3]^4+(-159567408/475*z^3)*x[0]^7*x[1]^6*x[3]^5+(-11243232/95*z^2)*x[0]^9*x[1]^3*x[2]*x[3]^5+(1061424/95*z^2)*x[1]^12*x[2]*x[3]^5+(-3164616/95*z)*x[0]^11*x[2]^2*x[3]^5+(5837832/95*z)*x[0]^2*x[1]^9*x[2]^2*x[3]^5+27243216/95*x[0]^4*x[1]^6*x[2]^3*x[3]^5+(-36324288/95*z^5+36324288/95*z^2)*x[0]^6*x[1]^3*x[2]^4*x[3]^5+(-51243192/475*z^4+51243192/475*z)*x[0]^8*x[2]^5*x[3]^5+(54486432/475*z^3-54486432/475)*x[0]*x[1]^6*x[2]^6*x[3]^5+(-36324288/95*z^5)*x[0]^3*x[1]^3*x[2]^7*x[3]^5+(-51243192/475*z^4)*x[0]^5*x[2]^8*x[3]^5+(9513504/475*z^2)*x[1]^3*x[2]^10*x[3]^5+(8923824/475*z)*x[0]^2*x[2]^11*x[3]^5-1092*x[0]^12*x[3]^6+5189184/95*x[0]^3*x[1]^9*x[3]^6+(-54486432/475*z^5+54486432/475*z^2)*x[0]^5*x[1]^6*x[2]*x[3]^6+(-44108064/95*z^4+44108064/95*z)*x[0]^7*x[1]^3*x[2]^2*x[3]^6+(240240/19*z^3-240240/19)*x[0]^9*x[2]^3*x[3]^6+(648648/95*z^3-648648/95)*x[1]^9*x[2]^3*x[3]^6+(54486432/95*z^5)*x[0]^2*x[1]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t[18] := 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i in {4,6,12,18} do f[i] := subs(z=zeta, f[i]) od:
for i in {12,18,24,36} do s[i] := subs(z=zeta, s[i]) od:
for i in {12,18,24} do t[i] := subs(z=zeta, t[i]) od:J := [seq(f[i], i in {4,6,12,18}), seq(s[i], i in {12,18,24,36}), seq(t[i], i in {12,18,24})]:
min_a := a^2-3:
fI := [z, 0, 4*z^3, 12691704*z^5*a, (496*z^3)/319, 24*z^5*a, (246016*z^6)/101761, (122023936*z^9)/32461759, (184*z^3)/205, 24*z^5*a, (91264*z^6)/65395];L:=create_lode(J,fI,[seq(x[i],i=0..3)],[Dz,z],[a],[min_a],[],[]);with(DETools):
with(PDETools):M:=de2diffop(dchange(z=1-t, diffop2de(L,[Dz,z],y(z))),y(t),[Dz,z]);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUpZXF1YWxpdHlHRjQvJStleGVjdXRhYmxlR0Y0Ri8=