## Polynomial Numerical Linear Algebra package for Julia

Multi-functional package for solving all kinds of problems with multivariate polynomials in double precision.

Author: Kim Batselier

## 1. Basics

Only 1 monomial ordering is supported: the degree negative lex ordering. This graded ordering is defined for two n-variate monomials x^a and x^b as

a > b

if

|a| = \sum_i^n a_i > |b| = \sum_i^n b_i or

|a|=|b| and the leftmost nonzero entry of a-b is negative.

In this definition x^a stands for

x^a = x_1^a_1 * x_2^a_2 * ... * x_n^a_n,

where a is an n-tuple of nonnegative integers.

A system of s multivariate polynomials is represented by an s-by-2 Any Array. The first column elements are vectors containing the coefficients. The second column elements are matrices containing the corresponding exponents. For example, the system

f_1 = 5.3 x_1^2 + 9 x_2 x_3 -1

f_2 = 2 x_1^3 + .5 x_2^2 - 7.89 x_3 - 94

f_3 = x_1 - 2.13

is represented by

polysys[1,1]=[5.3,9,-1]

polysys[1,2]=[2 0 0;0 1 1;0 0 0]

polysys[2,1]=[2,.5,-7.89,-94]

polysys[2,2]=[3 0 0;0 2 0;0 0 1;0 0 0]

polysys[3,1]=[1,-2.13]

polysys[3,2]=[1 0 0;0 0 0]

## 2. Available functions

index = feti(exponent)

Converts an exponent to its corresponding linear index with respect to the degree negative lex ordering. If the exponent argument is a matrix of exponents, then feti() is applied to each row of the matrix.

exponent = fite(index,n)

Converts a linear index with respect to the degree negative lex ordering to the corresponding n-variate exponent.

d0 = getD0(polysys)

Returns the maximal total degree of the given polynomial system.

dorig = getDorig(polysys)

dorig[i] (i=1:s) contains the maximal total degree of the multivariate polynomial corresponding with polysys[i,1],polysys[i,2].

(q,p) = getMDim(polysys,d)

Returns the number of columns p and the number of rows q of the Macaulay matrix of degree d.

M = getM(polysys,d,d0...)

Returns the Macaulay matrix for the given polynomial system polysys at degree d. When the optional third argument d0 is also given, then only the columns required to enlarge M(d0) to M(d) are returned.