# genericLaurentPolynomials -- generic (Laurent) polynomials

## Synopsis

• Usage:
genericLaurentPolynomials A
• Inputs:
• A, , or a list of $n+1$ matrices $A_0,\ldots,A_n$ over $\mathbb{Z}$ and with $n$ rows to represent the exponent vectors of (Laurent) polynomials $f_0,\ldots,f_n$ in $n$ variables. For the dense case, one can pass just the sequence of degrees of $f_0,\ldots,f_n$.
• Optional inputs:
• CoefficientRing => ..., default value ZZ
• Outputs:
• the generic (Laurent) polynomials $f_0,\ldots,f_n$ in the ring $\mathbb{Z}[a_0,a_1,\ldots,b_0,b_1,\ldots][x_1,\ldots,x_n]$, involving only the monomials from $A$. (Note that, if all the exponents are nonnegative, then the ambient polynomial ring is taken without inverses of variables, so that $f_0,\ldots,f_n$ are ordinary polynomials.)

## Description

This method helps to construct special types of sparse resultants, see for instance denseResultant.

 i1 : M = (matrix{{2,3,4,5},{0,2,1,0}},matrix{{1,-1,0,2,3},{-2,0,-7,-1,0}},matrix{{-1,0,6},{-2,1,3}}) o1 = (| 2 3 4 5 |, | 1 -1 0 2 3 |, | -1 0 6 |) | 0 2 1 0 | | -2 0 -7 -1 0 | | -2 1 3 | o1 : Sequence i2 : genericLaurentPolynomials M 5 4 3 2 2 3 2 -1 -2 -7 o2 = (a x + a x x + a x x + a x , b x + b x x + b x x + b x + 3 1 2 1 2 1 1 2 0 1 4 1 3 1 2 2 1 2 1 2 ------------------------------------------------------------------------ -1 6 3 -1 -2 b x , c x x + c x + c x x ) 0 1 2 1 2 1 2 0 1 2 o2 : Sequence i3 : genericLaurentPolynomials (2,3,1) 2 2 3 2 2 3 o3 = (a x + a x x + a x + a x + a x + a , b x + b x x + b x x + b x 5 1 4 1 2 2 2 3 1 1 2 0 9 1 8 1 2 6 1 2 3 2 ------------------------------------------------------------------------ 2 2 + b x + b x x + b x + b x + b x + b , c x + c x + c ) 7 1 5 1 2 2 2 4 1 1 2 0 2 1 1 2 0 o3 : Sequence