IsIntegerPointOf(zpoly,v) returns true if v is a point of zpoly, false otherwise.

Note that the coordinates of v must be integer numbers.

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

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

$\mathrm{ineqs}≔\left[0\le -16+2y+z\,0\le -72+4x+4y+3z\,0\le 2y-z\,0\le -24+4x+4y-3z\,0\le -4x+4y+3z\,0\le 48-4x+4y-3z\,0\le 48-4x-4y+3z\,0\le 8-2y+z\,0\le -24+4x-4y+3z\,0\le 24-2y-z\,0\le 24+4x-4y-3z\,0\le 96-4x-4y-3z\right]$

$\mathrm{ineqs}≔\left[0\le -16+2y+z\,0\le -72+4x+4y+3z\,0\le 2y-z\,0\le -24+4x+4y-3z\,0\le -4x+4y+3z\,0\le 48-4x+4y-3z\,0\le 48-4x-4y+3z\,0\le 8-2y+z\,0\le -24+4x-4y+3z\,0\le 24-2y-z\,0\le 24+4x-4y-3z\,0\le 96-4x-4y-3z\right]$

$L≔\mathrm{Lattice}\left(\mathrm{Matrix}\left(\left[\left[1\,0\,2\right]\,\left[0\,-1\,1\right]\,\left[0\,0\,2\right]\right]\right)\,\mathrm{Vector}\left(\left[0\,0\,1\right]\right)\right)$

$L≔\mathrm{Lattice}\left(\left[\begin{array}{ccc}1& 0& 2\\ 0& \mathrm{-1}& 1\\ 0& 0& 2\end{array}\right]\,\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\right)$

$\mathrm{vars}≔\left[z\,x\,y\right]$

$\mathrm{vars}≔\left[z\,x\,y\right]$

$\mathrm{zp}≔\mathrm{ZPolyhedralSet}\left(\mathrm{ineqs}\,\mathrm{vars}\,\mathrm{lattice}=L\right)$

$\mathrm{zp}≔\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}0\le 2y-z\\ 0\le -16+2y+z\\ 0\le 8-2y+z\\ 0\le 24-2y-z\\ 0\le -4x+4y+3z\\ 0\le -72+4x+4y+3z\\ 0\le -24+4x-4y+3z\\ 0\le -24+4x+4y-3z\\ 0\le 24+4x-4y-3z\\ 0\le 48-4x-4y+3z\\ 0\le 48-4x+4y-3z\\ 0\le 96-4x-4y-3z\end{array}\right.\\ \text{Variables}& \:& \left[z\,x\,y\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{ccc}1& 0& 2\\ 0& \mathrm{-1}& 1\\ 0& 0& 2\end{array}\right]\,\,\,\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\right)\end{array}\right.$

$\mathrm{Point\_from\_pzp}≔\mathrm{SamplePoint}\left(\mathrm{zp}\right)$

$\mathrm{Point\_from\_pzp}≔\left[z=10\,x=9\,y=7\right]$

$\mathrm{vect}≔\mathrm{Vector}\left(\left[\mathrm{seq}\left(\mathrm{rhs}\left(\mathrm{pt}\right)\,\mathrm{pt}=\mathrm{Point\_from\_pzp}\right)\right]\right)$

$\mathrm{vect}≔\left[\begin{array}{c}10\\ 9\\ 7\end{array}\right]$

$\mathrm{IsIntegerPointOf}\left(\mathrm{zp}\,\mathrm{vect}\right)$

$\mathrm{true}$

$\mathrm{vect}≔\mathrm{Vector}\left(\left[0\,0\,0\right]\right)$

$\mathrm{vect}≔\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

$\mathrm{IsIntegerPointOf}\left(\mathrm{zp}\,\mathrm{vect}\right)$

$\mathrm{false}$

$\mathrm{EnumerateIntegerPoints}\left(\mathrm{zp}\right)$

$\left[\left[x=9\,y=5\,z=6\right]\,\left[x=9\,y=7\,z=6\right]\,\left[x=8\,y=5\,z=7\right]\,\left[x=9\,y=5\,z=7\right]\,\left[x=10\,y=5\,z=7\right]\,\left[x=8\,y=7\,z=7\right]\,\left[x=9\,y=7\,z=7\right]\,\left[x=10\,y=7\,z=7\right]\,\left[x=7\,y=5\,z=8\right]\,\left[x=8\,y=5\,z=8\right]\,\left[x=9\,y=5\,z=8\right]\,\left[x=10\,y=5\,z=8\right]\,\left[x=11\,y=5\,z=8\right]\,\left[x=7\,y=7\,z=8\right]\,\left[x=8\,y=7\,z=8\right]\,\left[x=9\,y=7\,z=8\right]\,\left[x=10\,y=7\,z=8\right]\,\left[x=11\,y=7\,z=8\right]\,\left[x=8\,y=5\,z=9\right]\,\left[x=9\,y=5\,z=9\right]\,\left[x=10\,y=5\,z=9\right]\,\left[x=8\,y=7\,z=9\right]\,\left[x=9\,y=7\,z=9\right]\,\left[x=10\,y=7\,z=9\right]\,\left[x=9\,y=5\,z=10\right]\,\left[x=9\,y=7\,z=10\right]\right]$

Rachid Seghir, Vincent Loechner, and Benoı̂t Meister. "Integer affine transformations of parametric Z-polytopes and applications to loop nest optimization." Proceedings of TACO, Vol. 9(2):8:1–8:27, 2012.

Rui-Juan Jing and Marc Moreno Maza. "Computing the Integer Points of a Polyhedron, I: Algorithm." Proceedings of CASC 2017: 225-241, Springer.

Rui-Juan Jing and Marc Moreno Maza. "Computing the Integer Points of a Polyhedron, II: Complexity Estimates." Proceedings of CASC 2017: 242-256, Springer.

The PolyhedralSets:-ZPolyhedralSets:-IsIntegerPointOf command was introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.