# map(CoincidentRootLocus) -- the map associated to a coincident root locus

## Synopsis

• Function: map
• Usage:
map X
• Inputs:
• Optional inputs:
• Outputs:
• , a unirational parameterization of X

## Description

If $X\subset\mathbb{P}^n$ is the coincident root locus associated with the partition $(\lambda_1,\ldots,\lambda_d)$, then we have a map $\mathbb{P}^1\times\cdots\times\mathbb{P}^1\to\mathbb{P}^n$ which sends a $d$-tuple of linear forms $(L_1,\ldots,L_d)$ to ${L_1}^{\lambda_1}\cdots{L_d}^{\lambda_d}$. It is this map that is returned.

 i1 : X = coincidentRootLocus {3,2,2} o1 = CRL(3,2,2) o1 : Coincident root locus i2 : f = map X o2 = -- rational map -- source: Proj(QQ[t0 , t0 ]) x Proj(QQ[t1 , t1 ]) x Proj(QQ[t2 , t2 ]) 0 1 0 1 0 1 target: Proj(QQ[t , t , t , t , t , t , t , t ]) 0 1 2 3 4 5 6 7 defining forms: { 3 2 2 t0 t1 t2 , 0 0 0 2 3 2 2 3 2 3 2 2 2 -t0 t1 t2 t2 + -t0 t1 t1 t2 + -t0 t0 t1 t2 , 7 0 0 0 1 7 0 0 1 0 7 0 1 0 0 1 3 2 2 4 3 1 3 2 2 2 2 2 2 2 2 1 2 2 2 --t0 t1 t2 + --t0 t1 t1 t2 t2 + --t0 t1 t2 + -t0 t0 t1 t2 t2 + -t0 t0 t1 t1 t2 + -t0 t0 t1 t2 , 21 0 0 1 21 0 0 1 0 1 21 0 1 0 7 0 1 0 0 1 7 0 1 0 1 0 7 0 1 0 0 2 3 2 2 3 2 3 2 2 2 12 2 3 2 2 2 6 2 2 6 2 2 1 3 2 2 --t0 t1 t1 t2 + --t0 t1 t2 t2 + --t0 t0 t1 t2 + --t0 t0 t1 t1 t2 t2 + --t0 t0 t1 t2 + --t0 t0 t1 t2 t2 + --t0 t0 t1 t1 t2 + --t0 t1 t2 , 35 0 0 1 1 35 0 1 0 1 35 0 1 0 1 35 0 1 0 1 0 1 35 0 1 1 0 35 0 1 0 0 1 35 0 1 0 1 0 35 1 0 0 1 3 2 2 6 2 2 6 2 2 3 2 2 2 12 2 3 2 2 2 2 3 2 2 3 2 --t0 t1 t2 + --t0 t0 t1 t1 t2 + --t0 t0 t1 t2 t2 + --t0 t0 t1 t2 + --t0 t0 t1 t1 t2 t2 + --t0 t0 t1 t2 + --t0 t1 t2 t2 + --t0 t1 t1 t2 , 35 0 1 1 35 0 1 0 1 1 35 0 1 1 0 1 35 0 1 0 1 35 0 1 0 1 0 1 35 0 1 1 0 35 1 0 0 1 35 1 0 1 0 1 2 2 2 2 2 2 2 2 2 1 3 2 2 4 3 1 3 2 2 -t0 t0 t1 t2 + -t0 t0 t1 t1 t2 + -t0 t0 t1 t2 t2 + --t0 t1 t2 + --t0 t1 t1 t2 t2 + --t0 t1 t2 , 7 0 1 1 1 7 0 1 0 1 1 7 0 1 1 0 1 21 1 0 1 21 1 0 1 0 1 21 1 1 0 3 2 2 2 2 3 2 2 3 2 -t0 t0 t1 t2 + -t0 t1 t1 t2 + -t0 t1 t2 t2 , 7 0 1 1 1 7 1 0 1 1 7 1 1 0 1 3 2 2 t0 t1 t2 1 1 1 } o2 : MultihomogeneousRationalMap (rational map from PP^1 x PP^1 x PP^1 to PP^7) i3 : describe f o3 = rational map defined by multiforms of degree {3, 2, 2} source variety: PP^1 x PP^1 x PP^1 target variety: PP^7 image: 3-dimensional variety of degree 36 in PP^7 cut out by 364 hypersurfaces of degree 6 dominance: false birationality: false coefficient ring: QQ