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# genericPolynomials -- generic homogeneous polynomials

## Synopsis

• Usage:
genericPolynomials(d,K)
genericPolynomials d
• Inputs:
• d, a list, n+1 integers d_0,...,d_n
• K, a ring, optional, with default value QQ
• Outputs:
• a list, n+1 generic homogeneous polynomials of degrees d_0,...,d_n in the ring K[a_0,a_1,...,b_0,b_1,...][x_0,...,x_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 : genericPolynomials {1,2,3} 2 2 2 3 o1 = {a x + a x + a x , b x + b x x + b x + b x x + b x x + b x , c x 0 0 1 1 2 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 ------------------------------------------------------------------------ 2 2 3 2 2 2 2 + c x x + c x x + c x + c x x + c x x x + c x x + c x x + c x x 1 0 1 3 0 1 6 1 2 0 2 4 0 1 2 7 1 2 5 0 2 8 1 2 ------------------------------------------------------------------------ 3 + c x } 9 2 o1 : List i2 : first genericPolynomials({4,2,3},ZZ/101) 4 3 2 2 3 4 3 2 2 o2 = a x + a x x + a x x + a x x + a x + a x x + a x x x + a x x x + 0 0 1 0 1 3 0 1 6 0 1 10 1 2 0 2 4 0 1 2 7 0 1 2 ------------------------------------------------------------------------ 3 2 2 2 2 2 3 3 4 a x x + a x x + a x x x + a x x + a x x + a x x + a x 11 1 2 5 0 2 8 0 1 2 12 1 2 9 0 2 13 1 2 14 2 ZZ o2 : ---[a ..a , b ..b , c ..c ][x ..x ] 101 0 14 0 5 0 9 0 2 i3 : first genericPolynomials({4,-1,-1},ZZ/101) 4 3 2 2 3 4 3 2 2 o3 = a x + a x x + a x x + a x x + a x + a x x + a x x x + a x x x + 0 0 1 0 1 3 0 1 6 0 1 10 1 2 0 2 4 0 1 2 7 0 1 2 ------------------------------------------------------------------------ 3 2 2 2 2 2 3 3 4 a x x + a x x + a x x x + a x x + a x x + a x x + a x 11 1 2 5 0 2 8 0 1 2 12 1 2 9 0 2 13 1 2 14 2 ZZ o3 : ---[a ..a ][x ..x ] 101 0 14 0 2