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I have a problem for which I have to find binomials over a multivariate polynomial Ring which all have a common zero.

Let $\mathbb{F}[x_1,\dots,x_n]$ be some multivariate polynomial ring over some field.

Then I would like to find or generate (even better) a set of binomials (2-nomials) such that all of them have a common zero i.e.:

$f_1(\sigma) = \dots = f_r(\sigma)=0 $ where $\sigma \in \mathbb{F}^n$

So far I came up with binomials which all have a common zero in zero:

$xyz-x^3$ and similar constructions. But I need more and "better" ones.

When doing a change of coordinates from those, I do not necessarily end up with binomials.

Anybody ideas? Would be happy to get pointed to something as the term binomial doesn't gives the results I want from Google or Stackexchange.

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  • $\begingroup$ Your given example also generalises to a common zero with $\sigma = 1^n$. Is that "better"? $\endgroup$
    – Peter Taylor
    Commented Nov 23, 2021 at 14:08
  • $\begingroup$ No, I need to run some tests with them and would therefore like them to have just one common zero. $\endgroup$
    – Christoph Strobl
    Commented Nov 24, 2021 at 9:21

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