# tVeroneseSet -- give the Veronese set of t-spread monomials of a given degree

## Synopsis

• Usage:
tVeroneseSet(S,d,t)
• Inputs:
• S, a polynomial ring
• d, a nonnegative integer that identifies the degree of the monomials
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• a list, the set of all the t-spread monomials of S of degree d

## Description

the function tVeroneseSet(S,d,t) gives the Veronese set of t-spread monomials of degree d, that is, the set of all the t-spread monomials of S of degree dThis function calls the method tLexSeg(v,u,t), where v is the greatest t-spread monomial of the polynomial ring and u the smallest one. If $S=K[x_1,\ldots,x_n]$ then $v=x_1x_{1+t}\cdots x_{1+(d-1)t}$ and $u=x_{n-(d-1)t}\cdots x_{n-t} x_{n}$. This set is also denoted by $M_{n,d,t}$.

Examples:

 i1 : S=QQ[x_1..x_8] o1 = S o1 : PolynomialRing i2 : tVeroneseSet(S,3,2) o2 = {x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , 1 3 5 1 3 6 1 3 7 1 3 8 1 4 6 1 4 7 1 4 8 1 5 7 1 5 8 ------------------------------------------------------------------------ x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , 1 6 8 2 4 6 2 4 7 2 4 8 2 5 7 2 5 8 2 6 8 3 5 7 3 5 8 ------------------------------------------------------------------------ x x x , x x x } 3 6 8 4 6 8 o2 : List i3 : tVeroneseSet(S,3,3) o3 = {x x x , x x x , x x x , x x x } 1 4 7 1 4 8 1 5 8 2 5 8 o3 : List

• tNextMon -- give the t-lex successor of a given t-spread monomial
• tLexSeg -- give the t-lex segment with given extremes
• isTLexSeg -- whether a set of t-spread monomials is a t-lex segment
• tLexMon -- give the smallest initial t-lex segment containing a given monomial
• tVeroneseIdeal -- give the Veronese ideal of t-spread monomials of a given degree

## Ways to use tVeroneseSet :

• tVeroneseSet(Ring,ZZ,ZZ)

## For the programmer

The object tVeroneseSet is .