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I have a set of 20 multivariate polynomials in 5 dimensions $\big(f_i(x_1,x_2,x_3,x_4,x_5)=0\big)$. They are all 6th degree in each dimension. I am looking for a method to sample the intersection of these polynomials over some finite real region $(x_j \in [x_{j,min}, x_{j,max}])$.

The polynomials are Legendre basis polynomials which I interpolated over numerical simulations. This interpolation gives me a system of 20 polynomials $h_i(x_1,x_2,x_3,x_4,x_5)=g_i$ where $g_i$ are some scalars that I measure from an experiment. My goal is to get some sense of the possible combinations of $x_{1-5}$ that could have given me the measured result (for clarity: $f_i(x) = h_i(x) - g_i$)

While I can't prove it, I highly suspect that the intersection of all these polynomials is not finite but some hyper-surface. I know this to be the case if I only look at the first couple of polynomials; the idea behind using more is to try and narrow the size of the intersection. If this is true then I have a system of polynomials that appears to be positive-dimensional (per Wikipedia). Since I have 5 dimensions but 20 polynomials I don't think I can use Newton's method (with multiple starting points). I am hoping for suggestions on two points.

  1. Is there a simple way to describe the intersection of these functions $(\bigcap^{20}_{i=0} f_i(x_1,x_2,x_3,x_4,x_5) = 0)$? If there are just two polynomials I can just set them equal to each other; how do I generalize that to multiple polynomials? If I can analytically define the region then sampling it becomes easier (I think).

  2. If #1 is not straightforward, is there a good way to sample the region? My first thought is to define an objective function of the L2 norm of all the polynomials and try and minimize it using the Nelder–Mead method and use multiple starting points.

I am trying to get a sense of two things. The size of this region (some integral measure would suffice if #1 is possible) and also some idea of where this region is located or if there are multiple regions. Ideally any method would also allow me to look in a region near this intersection to account for noise in the experimental measurement.

I am aware of Gröbner bases, however, most of what I have read about them seems to focus on determining if there are finite roots, which I doubt to be the case here. I am an engineer, not a mathematician, so if I am missing something elementary please excuse my ignorance.

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