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# veronese -- Veronese embedding

## Synopsis

• Usage:
veronese(n,d,K)
veronese(n,d)
• Inputs:
• Optional inputs:
• Variable => ..., default value null, specify a name for a variable
• Outputs:
• , representing the $d$-th Veronese embedding of $\mathbb{P}^n$

## Description

This is an auxiliary method to build tests and examples. For instance, the two following codes have to produce the same polynomial up to a renaming of variables: 1) resultant genericPolynomials((n+1):d,K) and 2) fromPluckerToStiefel dualize chowForm veronese(n,d,K).

 i1 : veronese(1,4) 4 3 2 2 3 4 o1 = map (QQ[t ..t ], QQ[x ..x ], {t , t t , t t , t t , t }) 0 1 0 4 0 0 1 0 1 0 1 1 o1 : RingMap QQ[t ..t ] <-- QQ[x ..x ] 0 1 0 4 i2 : veronese(1,4,Variable=>y) 4 3 2 2 3 4 o2 = map (QQ[y ..y ], QQ[y ..y ], {y , y y , y y , y y , y }) 0 1 0 4 0 0 1 0 1 0 1 1 o2 : RingMap QQ[y ..y ] <-- QQ[y ..y ] 0 1 0 4 i3 : veronese(1,4,Variable=>(u,z)) 4 3 2 2 3 4 o3 = map (QQ[u ..u ], QQ[z ..z ], {u , u u , u u , u u , u }) 0 1 0 4 0 0 1 0 1 0 1 1 o3 : RingMap QQ[u ..u ] <-- QQ[z ..z ] 0 1 0 4 i4 : veronese(2,2,ZZ/101) ZZ ZZ 2 2 2 o4 = map (---[t ..t ], ---[x ..x ], {t , t t , t t , t , t t , t }) 101 0 2 101 0 5 0 0 1 0 2 1 1 2 2 ZZ ZZ o4 : RingMap ---[t ..t ] <-- ---[x ..x ] 101 0 2 101 0 5