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If you want x0 and y0 to be coefficients of your polynomials, your polynomial ring can not be defined over the ring QQ, but it should be defined over SR (the symbolic ring). So you can try something like:

sage: x0,y0 = var('x0,y0')
sage: R.<x1,y1,x2,y2,x3,y3> = PolynomialRing(SR, order='lex')
sage: sage: f1 = 13*(x1+x1*x2-y1*y2+(x1*x2-y1*y2)*x3-(y1*x2+y2*x1)*y3)-x0
sage: sage: f2 = 13*(y1+y1*x2+y2*x1+(y1*x2+y2*x1)*x3+(x1*x2-y1*y2)*y3)-y0
sage: sage: f3 = x1^2+y1^2-1
sage: sage: f4 = x2^2+y2^2-1
sage: sage: f5 = x3^2+y3^2-1
sage: I =R.ideal(f1,f2,f3,f4,f5)

Which results in

sage: I.groebner_basis()
verbose 0 (3369: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.

You can wait to see if something happens, but i am not very confident about how Sage will deal with the symbolic ring as it is currently implemented. A safer place to work in is in a well defined mathematical object, such as a polynomial ring. So, instead of claiming that x0 and y0 are symbols, let them be variables of a polynomial ring. Even better, since Groebner basis algorithms are more adapted to polynomials defined over a field, than over a ring, let us define x0 and y0 as variables of a fraction field:

sage: A.<x0,y0> = PolynomialRing(QQ)
sage: F = A.fraction_field()
sage: F.inject_variables()       # we redefine x0 and y0 in the larger field F
Defining x0, y0
sage: R.<x1,y1,x2,y2,x3,y3> = PolynomialRing(F, order='lex')
sage: R
Multivariate Polynomial Ring in x1, y1, x2, y2, x3, y3 over Fraction Field of Multivariate Polynomial Ring in x0, y0 over Rational Field
sage: I = R.ideal(f1,f2,f3,f4,f5)
sage: I.groebner_basis()
[x1 + (228488*y0/(26*x0^4 + 52*x0^2*y0^2 + 26*y0^4 - 4394*x0^2 - 4394*y0^2))*y2^2*y3 + ((-676*y0)/(26*x0^2 + 26*y0^2))*y2*x3 + ((-228488*y0)/(26*x0^4 + 52*x0^2*y0^2 + 26*y0^4 - 4394*x0^2 - 4394*y0^2))*y2*y3^2 + (338*x0/(26*x0^2 + 26*y0^2))*x3 + ((-338*y0)/(26*x0^2 + 26*y0^2))*y3 + (-x0^3 - x0*y0^2 + 169*x0)/(26*x0^2 + 26*y0^2), y1 + ((-228488*x0)/(26*x0^4 + 52*x0^2*y0^2 + 26*y0^4 - 4394*x0^2 - 4394*y0^2))*y2^2*y3 + (676*x0/(26*x0^2 + 26*y0^2))*y2*x3 + (228488*x0/(26*x0^4 + 52*x0^2*y0^2 + 26*y0^4 - 4394*x0^2 - 4394*y0^2))*y2*y3^2 + (338*y0/(26*x0^2 + 26*y0^2))*x3 + (338*x0/(26*x0^2 + 26*y0^2))*y3 + (-x0^2*y0 - y0^3 + 169*y0)/(26*x0^2 + 26*y0^2), x2 + ((-77228944)/(338*x0^4 + 676*x0^2*y0^2 + 338*y0^4 - 114244*x0^2 - 114244*y0^2 + 9653618))*y2^3*y3 + ((-228488)/(338*x0^2 + 338*y0^2 - 57122))*y2^2 + (456976/(338*x0^2 + 338*y0^2 - 57122))*y2*x3*y3 + (77228944/(338*x0^4 + 676*x0^2*y0^2 + 338*y0^4 - 114244*x0^2 - 114244*y0^2 + 9653618))*y2*y3^3 + (-2)*y2*y3 + 3*x3 + (228488/(338*x0^2 + 338*y0^2 - 57122))*y3^2 - 1/338*x0^2 - 1/338*y0^2 + 1/2, y2^2*x3 + y2^2 - y2*x3*y3 + (1/338*x0^2 + 1/338*y0^2 - 3/2)*y2*y3 + (-1/338*x0^2 - 1/338*y0^2 + 1/2)*x3 + 1/228488*x0^4 + 1/114244*x0^2*y0^2 + 1/228488*y0^4 - 3/676*x0^2 - 3/676*y0^2 + 5/8, y2^2*y3^2 + (-1/338*x0^2 - 1/338*y0^2 + 1/2)*y2*x3*y3 - y2*y3^3 + (1/338*x0^2 + 1/338*y0^2 - 1/2)*y2*y3 + (-1/228488*x0^4 - 1/114244*x0^2*y0^2 - 1/228488*y0^4 + 1/676*x0^2 + 1/676*y0^2 - 1/8)*x3 + (-1/338*x0^2 - 1/338*y0^2 + 1/2)*y3^2 + 1/228488*x0^4 + 1/114244*x0^2*y0^2 + 1/228488*y0^4 - 1/676*x0^2 - 1/676*y0^2 + 1/8, x3^2 + y3^2 - 1]

If you want x0 and y0 to be coefficients of your polynomials, your polynomial ring can not be defined over the ring QQ, but it should (in a first attempt) be defined over SR (the symbolic ring). So you can try something like:

sage: x0,y0 = var('x0,y0')
sage: R.<x1,y1,x2,y2,x3,y3> = PolynomialRing(SR, order='lex')
sage: sage: f1 = 13*(x1+x1*x2-y1*y2+(x1*x2-y1*y2)*x3-(y1*x2+y2*x1)*y3)-x0
sage: sage: f2 = 13*(y1+y1*x2+y2*x1+(y1*x2+y2*x1)*x3+(x1*x2-y1*y2)*y3)-y0
sage: sage: f3 = x1^2+y1^2-1
sage: sage: f4 = x2^2+y2^2-1
sage: sage: f5 = x3^2+y3^2-1
sage: I =R.ideal(f1,f2,f3,f4,f5)

Which results in

sage: I.groebner_basis()
verbose 0 (3369: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.

You can wait to see if something happens, but i am not very confident about how Sage will deal with the symbolic ring as it is currently implemented. implemented.

A safer place to work in is in a well defined mathematical object, such as a polynomial ring. So, instead of claiming that x0 and y0 are symbols, let them be variables undeterminates of a polynomial ring. Even better, since Groebner basis algorithms are more adapted to polynomials defined over a field, than over a ring, let us define x0 and y0 as variables undeterminates of a fraction field:

sage: A.<x0,y0> = PolynomialRing(QQ)
sage: F = A.fraction_field()
sage: F.inject_variables()       # we redefine x0 and y0 in the larger field F
Defining x0, y0
sage: R.<x1,y1,x2,y2,x3,y3> = PolynomialRing(F, order='lex')
sage: R
Multivariate Polynomial Ring in x1, y1, x2, y2, x3, y3 over Fraction Field of Multivariate Polynomial Ring in x0, y0 over Rational Field
sage: I = R.ideal(f1,f2,f3,f4,f5)
sage: I.groebner_basis()
[x1 + (228488*y0/(26*x0^4 + 52*x0^2*y0^2 + 26*y0^4 - 4394*x0^2 - 4394*y0^2))*y2^2*y3 + ((-676*y0)/(26*x0^2 + 26*y0^2))*y2*x3 + ((-228488*y0)/(26*x0^4 + 52*x0^2*y0^2 + 26*y0^4 - 4394*x0^2 - 4394*y0^2))*y2*y3^2 + (338*x0/(26*x0^2 + 26*y0^2))*x3 + ((-338*y0)/(26*x0^2 + 26*y0^2))*y3 + (-x0^3 - x0*y0^2 + 169*x0)/(26*x0^2 + 26*y0^2), y1 + ((-228488*x0)/(26*x0^4 + 52*x0^2*y0^2 + 26*y0^4 - 4394*x0^2 - 4394*y0^2))*y2^2*y3 + (676*x0/(26*x0^2 + 26*y0^2))*y2*x3 + (228488*x0/(26*x0^4 + 52*x0^2*y0^2 + 26*y0^4 - 4394*x0^2 - 4394*y0^2))*y2*y3^2 + (338*y0/(26*x0^2 + 26*y0^2))*x3 + (338*x0/(26*x0^2 + 26*y0^2))*y3 + (-x0^2*y0 - y0^3 + 169*y0)/(26*x0^2 + 26*y0^2), x2 + ((-77228944)/(338*x0^4 + 676*x0^2*y0^2 + 338*y0^4 - 114244*x0^2 - 114244*y0^2 + 9653618))*y2^3*y3 + ((-228488)/(338*x0^2 + 338*y0^2 - 57122))*y2^2 + (456976/(338*x0^2 + 338*y0^2 - 57122))*y2*x3*y3 + (77228944/(338*x0^4 + 676*x0^2*y0^2 + 338*y0^4 - 114244*x0^2 - 114244*y0^2 + 9653618))*y2*y3^3 + (-2)*y2*y3 + 3*x3 + (228488/(338*x0^2 + 338*y0^2 - 57122))*y3^2 - 1/338*x0^2 - 1/338*y0^2 + 1/2, y2^2*x3 + y2^2 - y2*x3*y3 + (1/338*x0^2 + 1/338*y0^2 - 3/2)*y2*y3 + (-1/338*x0^2 - 1/338*y0^2 + 1/2)*x3 + 1/228488*x0^4 + 1/114244*x0^2*y0^2 + 1/228488*y0^4 - 3/676*x0^2 - 3/676*y0^2 + 5/8, y2^2*y3^2 + (-1/338*x0^2 - 1/338*y0^2 + 1/2)*y2*x3*y3 - y2*y3^3 + (1/338*x0^2 + 1/338*y0^2 - 1/2)*y2*y3 + (-1/228488*x0^4 - 1/114244*x0^2*y0^2 - 1/228488*y0^4 + 1/676*x0^2 + 1/676*y0^2 - 1/8)*x3 + (-1/338*x0^2 - 1/338*y0^2 + 1/2)*y3^2 + 1/228488*x0^4 + 1/114244*x0^2*y0^2 + 1/228488*y0^4 - 1/676*x0^2 - 1/676*y0^2 + 1/8, x3^2 + y3^2 - 1]