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Consider the ideal $$I=\left\{p_2 \left(p_3^2-p_4^2\right) \left(p_5 p_6+p_7\right),-\left(p_2 p_3-p_1 p_4\right) \left(p_5 p_6+p_7\right),-\left(p_3+p_4\right) \right\}.$$

Corollary 1.3 gives a technique to use elimination ideal repeatedly to test whether the ideal is binomial. Macaulay2 has the Binomials package but some unexpected results here.

Is there a command to test whether an ideal is binomial in Mathematica?

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    `In[111]:= ii = {p2(p3^2 - p4^2)(p5p6 + p7), (p2p3 - p1p4)(p5*p6 + p7), p3 + p4}; gb = GroebnerBasis[ii]

    Out[112]= {p3 + p4, p1 p4 p5 p6 + p2 p4 p5 p6 + p1 p4 p7 + p2 p4 p7}` That should tell you it does not have exclusively binomial generators

     – Daniel Lichtblau Apr 11 '16 at 21:19

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