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My system has three equations and three unknowns, which I want to transform into a series of three 1-D root finding problems.

For this to be possible, the system must have a finite number of solutions, but your system has an infinite number of solutions.

is there a test to see if my system of equations is pathological in some way?

You can compute I.dimension(). For 0-dimensional ideals (like the Scholarpedia example), the output will be more like what you expect.

My system has three equations and three unknowns, which I want to transform into a series of three 1-D root finding problems.

For this to be possible, possible the system must have a finite number of solutions, but your system has an infinite number of solutions.

is there a test to see if my system of equations is pathological in some way?

You can compute I.dimension(). (By the way, this uses the Gröbner basis.) For 0-dimensional ideals (like the Scholarpedia example), example) there are finitely many solutions over $\mathbb{C}$, and the output will be more like what you expect.

My system has three equations and three unknowns, which I want to transform into a series of three 1-D root finding problems.

For this to be possible the system must have a finite number of solutions, but your system has an infinite number of solutions.

is there a test to see if my system of equations is pathological in some way?

You can compute I.dimension(). (By the way, this uses the Gröbner basis.) For 0-dimensional ideals in $\mathbb{Q}[x_1,\ldots,x_n]$ (like the Scholarpedia example) there are finitely many solutions over $\mathbb{C}$, and the output will be more like what you expect.

My system has three equations and three unknowns, which I want to transform into a series of three 1-D root finding problems.

For this to be possible the system must have a finite number of solutions, but your system has an infinite number of solutions.

is there a test to see if my system of equations is pathological in some way?

You can compute I.dimension(). (By the way, this uses the Gröbner basis.) For 0-dimensional ideals in $\mathbb{Q}[x_1,\ldots,x_n]$ (like the Scholarpedia example) there are finitely many solutions over $\mathbb{C}$, and the output will be what you expect.

My system has three equations and three unknowns, which I want to transform into a series of three 1-D root finding problems.

For this to be possible the system must have a finite number of solutions, but your system has an infinite number of solutions.

is there a test to see if my system of equations is pathological in some way?

You can compute I.dimension(). (By the way, this uses the Gröbner basis.) For 0-dimensional ideals in $\mathbb{Q}[x_1,\ldots,x_n]$ (like the Scholarpedia example) there are finitely many solutions over $\mathbb{C}$, and the output will be what you expect.

You can also call I.variety(ring=R) to solve the system of equations, equations over R (e.g. CC or RR for approximate solutions;AA or QQbar for exact solutions), which will give a warning if the ideal is not 0-dimensional.

My system has three equations and three unknowns, which I want to transform into a series of three 1-D root finding problems.

For this to be possible the system must have a finite number of solutions, but your system has an infinite number of solutions.

is there a test to see if my system of equations is pathological in some way?

You can compute I.dimension(). (By the way, this uses the Gröbner basis.) For 0-dimensional ideals in $\mathbb{Q}[x_1,\ldots,x_n]$ (like the Scholarpedia example) there are finitely many solutions over $\mathbb{C}$, and the output will be what you expect.

You can also call I.variety(ring=R) to solve the system of equations over R (e.g. CC or RR for approximate solutions;solutions, AA or QQbar for exact solutions), which will give a warning an error if the ideal is not 0-dimensional.