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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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hhh
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Another perspective on binomial ideals is provided by Combinatorial Commutative Algebra. This field explores algebraic structures arising from combinatorics, such as graphs, simplicial complexes, polytopes, and more.

My "bible" in this area is the book

Herzog, Jürgen; Hibi, Takayuki; Ohsugi, Hidefumi, Binomial ideals, Graduate Texts in Mathematics 279. Cham: Springer (ISBN 978-3-319-95347-2/hbk; 978-3-319-95349-6/ebook). xix, 321 p. (2018). ZBL1403.13004.

which offers a deep study of prime binomial ideals (i.e., toric ideals) and numerous applications in combinatorics and statistics.