I am studying ideals such as toric ideals but I am unable to find a consistent definition, it seems to be very general so please explain the origin of "toric ideal". Is there a geometric interpretation for toric ideal for example in three dimensions?

Has toric ideal something to do with torus?


In some sense yes, but the terminology is a bit confusing. Let me try to explain.

First of all, "torus" in this case doesn't refer the usual doughnut-like torus, but rather an algebraic torus, meaning an algebraic group of the form $(k^\times)^n$ for some field $k$. (When $k=\mathbf C$ this group is homotopy equivalent to the usual torus $(S^1)^n$, hence the name.)

Now, as you presumably know, there is a branch of algebraic geometry called toric geometry. This studies algebraic varieties obtained as equivariant partial compactifications of the algebraic torus $(k^\times)^n$.

It turns out that these varieties can be obtained by gluing affine varieties of a very special kind, namely ones whose ideal is generated by binomials. Here a binomial is an element in $k[x_1,\ldots,x_n]$ of the form $\prod_i x_i^{u_i}-\prod_jx_j^{v_j}$ for some exponent vectors $(u_1,\ldots,u_n)$ and $(v_1,\ldots,v_n)$.

For this reason, an ideal in $k[x_1,\ldots,x_n]$ generated by binomials is called a toric ideal.

  • $\begingroup$ Does this mean that multivariate terms such as $x_1x_3$ are not included in any of the sum? $\endgroup$ – hhh Apr 11 '16 at 11:51
  • 1
    $\begingroup$ Sorry, I made a typing error. Now fixed! $\endgroup$ – Nefertiti Apr 11 '16 at 12:59

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