# initialYForms -- computes the ideal of initial y-forms

## Synopsis

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

## 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.