Hi SageMath Inc developers,
First there is an open problem on mathoverflow at [1]:
Conjectured Somos-like closed form of recurrences with polynomial coefficients
I would like a proof if possible.
The answer of Max Alekseyev there links to code using Groebner basis
over QQ.
If I modify the attached code adding `.groebner_basis(deg_bound=2)`,
I get series of errors:
====
RuntimeError: error in Singular function call 'groebner':
overflow at t^20
error occurred in or before standard.lib::stdhilb line 291: `
intvec hi = hilb( Id[1],1,W );`
expected intvec-expression. type 'help intvec;'
leaving standard.lib::stdhilb (0)
leaving standard.lib::groebner (1128)
===
If I remove deg_bound=2, the code works.
Attached is the modified testcase.
[1]: https://mathoverflow.net/q/476226/12481
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# assume that y[i] = f(n-d+i)
dF1 = (2, lambda n,y: (3*n+5)*y[0] + (n^2 + n + 2)*y[1])
def find_fg(dF):
d, F = dF
k = 1
while True:
K = PolynomialRing(QQ,k+d+1+1,'y',order='invlex')
y = K.gens()[:-1]
n = K.gens()[-1]
Y = [ y[d+i] - F(n+i,y[i:i+d]) for i in range(k+1) ]
sol = [-g.coefficient({y[-1]:0})/g.coefficient({y[-1]:1}) for g in
K.ideal(Y).groebner_basis(deg_bound=2) if g.degree(n)==0 and g.degree(y[-1])==1]
if sol:
print(f'order = {k+d}')
# beautify presentation of solutions
KZ = PolynomialRing(ZZ,k+d,'y',order='invlex')
KZ._latex_names = [f'f(n-{k+d-i})' for i in range(k+d)]
KZF = KZ.fraction_field()
return ['f(n) = ' + latex(KZF(s)) for s in sol]
k += 1
find_fg(dF1)
