A$, and is the maximal degree of entries of ${\bm{v}}$ or $A$. If $A$ has no zero rows the [*row degrees*]{} of $A$, denoted by ${{\textnormal{rowdeg}}}\, A$, is the tuple $(d_1,\ldots,d_n)$ with $d_i = \deg
{{\textnormal{row}}}(A,i)$.

The (row-wise) [*leading matrix*]{} of $A$, denoted by ${\rm LM}(A) \in {{\mathsf{K}}}^{n \times m}$, has ${\rm LM}(A)_{i,j}$ equal to the coefficient of $x^{d_i}$ of $A_{i,j}$.

Next we recall [@barel_general_1992; @zhou_efficient_2012; @jeannerod_computation_2016] the shifted variants of the notion of degree, row degrees, and leading matrix. For a [*shift*]{} ${\bm{s}} =
(s_1,\ldots,s_n) \in {\mathbb Z\xspace}^n$, define the $n \times n$ diagonal matrix $x^{{\bm{s}}}$ by $$x^{{\bm{s}}} := \left [ \begin{array}{ccc} x^{s_1}
& & \\
 &  \ddots & \\ & & x^{s_n} \end{array} \right ].$$ Then the [*${{\bm{s}}}$-degree*]{} of $v$, the [*${{\bm{s}}}$-row degrees*]{} of $A$, and the [*${\bm{s}}$-leading matrix*]{} of $A$, are defined by $\deg_{{\bm{s}}} v := \deg v x^{{\bm{s}}}$, ${{\textnormal{rowdeg}}}_{{\bm{s}}} A := {{\textnormal{rowdeg}}}\, Ax^{{\bm{s}}}$, and ${\rm LM}_{{\bm{s}}}(A) := {\rm
LM}(Ax^{{\bm{s}}})$. Note that we pass over the ring of Laurent polynomials only for convenience; our algorithms will only compute with polynomials. As pointed out in [@jeannerod_computation_2016], up to negation the definition of ${{\bm{s}}}$-degree is equivalent to that used in [@BeckermannLabahnVillard06] and to the notion of [*defect*]{} in [@beckermann_uniform_1994].

