'canonical'=truefalse

The canonical option controls the redundancy of the output. Its default value is false. If 'canonical'=true, or just 'canonical' for short, is specified, it will be guaranteed that there are no redundancies among the constraints with the same main variables in the output.

'projection'=nonnegint

The projection option determines the subcoordinate space to be projected onto. If the option 'projection'=k is specified, where $k$ is a nonnegative integer less than the number of variables, then the output provides the projection of the zero set of $S$ onto the coordinate subspace determined by the smallest $k$ variables of R.

The algorithm behind this command is a variant of Fourier-Motzkin elimination.

Consider the linear semi-algebraic system $S$  whose unknowns are the variables of R and whose equations, non-negative inequalities, strictly positive inequalities, and inequations are given either by sys or by F, N, P, and H, respectively.

The commands LinearSolve(sys, R) and LinearSolve(F, N, P, H, R) return an equivalent linear semi-algebraic system of triangular shape to the input linear semi-algebraic system $S$. This assumes that R is a polynomial ring over the field of rational numbers.

If $S$ has no solutions, the output is the empty list $\left[\right]$. If the solution of $S$ is the whole space, the output is $\left[\mathrm{true}\right]$. Otherwise, the output is a nonempty list of linear constraints satisfying the following properties:

The output constraints are in an ascending order according to the largest variable appearing in them.

The projection of the solutions of the input system $S$ onto any lower dimensional space, say the space formed by the smallest $i$ variables, are exactly the solutions of those constraints in the output which only involve the smallest $i$ variables.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[z\,y\,x\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left[x+y+z=0\&comma;0\le x+y-z\&comma;x-y-z<0\right]$

$\mathrm{sys}≔\left[x+y+z=0\&comma;0\le x+y-z\&comma;x-y-z<0\right]$

$T≔\mathrm{LinearSolve}\left(\mathrm{sys}\&comma;R\right)$

$T≔\left[x<0\&comma;-x\le y\&comma;z=-x-y\right]$

$T≔\mathrm{LinearSolve}\left(\mathrm{sys}\&comma;R\&comma;'\mathrm{projection}'=2\right)$

$T≔\left[x<0\&comma;-x\le y\right]$

$\mathrm{sys}≔\left[x+y+z\le 1\&comma;x+y+z\le 2\&comma;0\le x+y-z\right]$

$\mathrm{sys}≔\left[z+y+x\le 1\&comma;z+y+x\le 2\&comma;0\le x+y-z\right]$

$T≔\mathrm{LinearSolve}\left(\mathrm{sys}\&comma;R\right)$

$T≔\left[z\le x+y\&comma;z\le 2-y-x\&comma;z\le 1-y-x\right]$

$T≔\mathrm{LinearSolve}\left(\mathrm{sys}\&comma;R\&comma;'\mathrm{canonical}'='\mathrm{true}'\right)$

$T≔\left[z\le x+y\&comma;z\le 1-y-x\right]$

The F, N, P, H and options parameters were introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.