initialYForms -- computes the ideal of initial y-forms

Synopsis

Usage:

initialYForms(I, y)
Inputs:
- I, an ideal,
- y, a ring element, an indeterminate in the ring
Optional inputs:
- UniversalGB => ..., default value false, whether the generators for an ideal form a universal Gröbner basis
Outputs:
- an ideal,

Description

Let $y$ be a variable of the polynomial ring $R = k[x_1,\ldots,x_n]$. A monomial ordering $<$ on $R$ is said to be $y$-compatible if the initial term of $f$ satisfies ${\rm in}_<(f) = {\rm in}_<({\rm in}_y(f))$ for all $f \in R$. Here, ${\rm in}_y(f)$ is the initial $y$-form of $f$, that is, if $f = \sum_i \alpha_iy^i$ and $\alpha_d \neq 0$ but $\alpha_t = 0$ for all $t >d$, then ${\rm in}_y(f) = \alpha_d y^d$. We set ${\rm in}_y(I) = \langle {\rm in}_y(f) ~|~ f \in I \rangle$ to be the ideal generated by all the initial $y$-forms in $I$

This routine computes the ideal of initial $y$-forms ${\rm in}_y(I)$.

For more on the definition of initial $y$-forms or their corresponding ideals, see [KMY, Section 2.1]. The following example is [KR, Example 2.16].

i1 : R = QQ[x,y,z,w,r,s]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(y*(z*s - x^2), y*w*r, w*r*(z^2 + z*x + w*r + s^2))

               2                             2       2 2        2
o2 = ideal (- x y + y*z*s, y*w*r, x*z*w*r + z w*r + w r  + w*r*s )

o2 : Ideal of R

i3 : initialYForms(I, y)

                       2       2 2        2          2
o3 = ideal (x*z*w*r + z w*r + w r  + w*r*s , y*w*r, x y - y*z*s)

o3 : Ideal of R

References

[KMY] Allen Knutson, Ezra Miller, and Alexander Yong. Gröbner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math. 630 (2009) 1–31.

[KR] Patricia Klein and Jenna Rajchgot. Geometric vertex decomposition and liaison. Forum Math. Sigma, 9 (2021) e70:1-23.

Ways to use initialYForms:

initialYForms(Ideal,RingElement)

For the programmer