# Gröbner base for some lexicographic order

• Nov 7th 2008, 08:04 PM
roporte
Gröbner base for some lexicographic order
Let $A(a_{ij})$ a non null real matrix of dimension $m \times n$. Let $I$ the ideal generated by the $m$ polynomials $\sum_i a_{ji}x_i$. Show that this polynomials are a Gröbner base of the ideal I for some lexicographic order.

Thanks!!
• Jan 13th 2015, 05:14 PM
Rebesques
Re: Gröbner base for some lexicographic order
Consider the lexicographic order $x_1.

Reduce the matrix in row echelon form using row operations; The polynomial set produces the same ideal, and cannot be reduced any further.
Thus, this set (and equivalently the starting set) is a Groebner basis.