# Finding relations between polynomials

Suppose I have a multivariate polynomial ring $A=\mathbb{R}[x_1,\ldots,x_n]$ and a set of $S=\{p_1, \ldots,p_k\}$ polynomials in $A$. Using this code (which works great) Dimension of an algebraic variety I found that if I consider a family of ideals $I_1=(p_1)$, $I_2=(p_1, p_2)$ etc. at certain point the Krull dimension does not change, ie. the dimension of $I_s=(p_1,\ldots,p_s)$ is the same as the dimension of $I_{s+1}=(p_1,\ldots,p_s,p_{s+1})$. It should mean that there is an algebraic relation between $p_1,\ldots,p_{s+1}$ and my question would be: how to extract it?

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Compute gb=GroebnerBasis[{p1,...,ps},vars,termorder] and then PolynomialReduce[psplus1,gb,vars,termorder]. –  Daniel Lichtblau Dec 9 '13 at 14:20