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I want to understand sum of binomials better in terms of ideals such as binomial ideals, normal ideals and so by toric ideals. Examples about toric ideals contain $$\sum x^\alpha+\sum x^\beta\in\mathbb R$$ where $\alpha,\beta$ are multidegrees. I have a gut feeling that toric varieties and this Sparse elimination theory is crucial to understand features of $\sum x^\alpha+\sum x^\beta$ deeper such as its negativity and positivity conditions. By ideal-variety correspondence, I think this question is related to Reference request: toric geometry while vaguely on Finding generators of toric ideals. Sturmfels considers toric ideals in $\mathbb C$ and acknowledges the fact that Cox preassumes toric ideals to be normal while the work uses some other definition. So

Does there exist a "bible" reference work on definitions on binomial ideals, normal ideals and then to toric ideals?

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Fulton may be one of the earliest publication on toric varieties. It contains analysis on lattice points in polytopes and toric varieties.

References

  • W. Fulton, Introduction to toric varieties, Annals of Math. Studies, Vol. 131, Princeton University Press, 1993.

Examples about applications

  • Toric ideals and toric ideals of graphs here

  • In the contex of contingency tables and markov bases such that $z=z^+-z^-$ so binomial $u^{z^+}-u^{z^-}$ where $u^{z^+}=\prod_i u(i)^{z^{+}(i)}$, $u^{z^-}=\prod_{i,j} u(i)^{z^-(i)}$ such that $z^+$ is the positive part while $z^-$ is the negative part -- source page 6 here. Toric ideal is defined as $I_A=\langle u^{z^+}-u^{z^-}; Az=0\rangle$.

Other threads

  1. Has toric ideal something to do with torus? where algebraic geometry, toric geometry and gluing binomials to get toric ideals
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