Hi sage devs,
I am proposing contribution to sage.
Attaching https://j.ludost.net/code/algdep_numeric.py.txt
>From the comments:
Given a list F of numeric lists F_{i,1},F_{i,2},...F_{i,k}
Finds a list of polynomials G_i, such that G_i(F_j)=0
Examples:
F=[ [ZZ(fibonacci(2*i)),fibonacci(2*i+1)] for i in range(2,50)]
ag=algdepanalytic(F,D=2,check=1);ag
[x0^2 + x0*x1 - x1^2 + 1]
The X coordinate of an Elliptic Curve over finite field:
F=[ X(n*P),X((n+1)*P)]
p=next_prime(10**3);K=GF(p)
E=EllipticCurve(GF(p),[0,2]);P=E.gens()[0]
def X_n(P,n): return (n*P).xy()[0]
F=[ [X_n(P,2*i),X_n(P,2*i+1)] for i in range(2,47)]
ag=algdepanalytic(F,D=4,check=1);ag
[126*x0^2*x1^2 - 97*x0^2*x1 - 97*x0*x1^2 + 353*x0^2 + 303*x0*x1 +
353*x1^2 + x0 + x1 - 388]
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def algdepanalytic(F,D=4,check=False,K=None):
"""
sage code for algebraic dependendecy of numerical data
Author: Georgi Guninski, ggunin...@gmail.com, gunin...@guninski.com
Version: 2.0, Mon 06 Feb 2023
Project page: https://j.ludost.net/code
Given a list F of numeric lists F_{i,1},F_{i,2},...F_{i,k}
Finds a list of polynomials G_i, such that G_i(F_j)=0
Examples:
FI=[fibonacci(2*i),fibonacci(2*i+1)]
def FI(n): return fibonacci(n)
F=[ [ZZ(fibonacci(2*i)),fibonacci(2*i+1)] for i in range(2,50)]
ag=algdepanalytic(F,D=2,check=1);ag
[x0^2 + x0*x1 - x1^2 + 1]
The X coordinate of an Elliptic Curve over finite field:
F=[ X(n*P),X((n+1)*P)]
p=next_prime(10**3);K=GF(p)
E=EllipticCurve(GF(p),[0,2]);P=E.gens()[0]
def X_n(P,n): return (n*P).xy()[0]
F=[ [X_n(P,2*i),X_n(P,2*i+1)] for i in range(2,47)]
ag=algdepanalytic(F,D=4,check=1);ag
[126*x0^2*x1^2 - 97*x0^2*x1 - 97*x0*x1^2 + 353*x0^2 + 303*x0*x1 + 353*x1^2 + x0 + x1 - 388]
"""
n=len(F[0])
if K is None: K=F[0][0].base_ring()
Kx=PolynomialRing(K,x,n)
ge=Kx.gens()
mo=prod(ge)**D
mo=Kx.monomial_all_divisors(mo)
mo=[i for i in mo if i.degree()<=D]
mo += [Kx(1)]
if len(F)<len(mo) and check:
print("underdetermined")
raise Exception(f"underdetermined and check=True need={len(mo)}")
J=[]
for P in F:
l=[i(P) for i in mo]
J.append(l)
M=Matrix(K,J)
ke=list(M.right_kernel().basis())
res=[]
for k in ke:
S=sum(i*j for i,j in zip(k,mo))
res += [S]
return res
