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Consider the following program

A.<a1, a2, a3, l2, l3> = PolynomialRing(QQ)
F = A.fraction_field()
F.inject_variables()
R.<c2, s2, c1, s1> = PolynomialRing(F, order = 'lex')
I = R.ideal(l3*s1*s2 + l2*c1 - a1, l3*s1*c2 - l2*s1 - a2, l3*c2 - a3, c1^2 + s1^2 - 1, c2^2 + s2^2 - 1)
I.groebner_basis()

So $a_1, a_2, a_3, l_2, l_3$ are parameters, and I am trying to solve the system of equations

$$a_1 = l_3s_1s_2 + l_2c_1 $$

$$a_2 = l_3s_1c_2 - l_2s_1$$

$$a_3 = l_3c_2$$

$$c_1^2 + s_1^1 = 1$$

$$c_2^2 + s_2^2 = 2$$

Now this gives as result 1, meaning that there is no solution to this system. However, for $a_1 = l_2$, $a_2 = 0$ and $a_3 = l_3$ we do have a solution, namely $c_1 = 1, s_1 = 0, c_2 = 1, s_2 = 0$.

So is something in the programming going wrong, or am I misunderstanding something on the algebraic geometry part?

Consider the following program

A.<a1, a2, a3, l2, l3> = PolynomialRing(QQ)
F = A.fraction_field()
F.inject_variables()
R.<c2, s2, c1, s1> = PolynomialRing(F, order = 'lex')
I = R.ideal(l3*s1*s2 + l2*c1 - a1, l3*s1*c2 - l2*s1 - a2, l3*c2 - a3, c1^2 + s1^2 - 1, c2^2 + s2^2 - 1)
I.groebner_basis()

So $a_1, a_2, a_3, l_2, l_3$ are parameters, and I am trying to solve the system of equations

$$a_1 = l_3s_1s_2 + l_2c_1 $$

$$a_2 = l_3s_1c_2 - l_2s_1$$

$$a_3 = l_3c_2$$

$$c_1^2 + s_1^1 = 1$$

$$c_2^2 + s_2^2 = 2$$1$$

Now this gives as result 1, meaning that there is no solution to this system. However, for $a_1 = l_2$, $a_2 = 0$ and $a_3 = l_3$ we do have a solution, namely $c_1 = 1, s_1 = 0, c_2 = 1, s_2 = 0$.

So is something in the programming going wrong, or am I misunderstanding something on the algebraic geometry part? part?