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RingChoice -- optional argument for polyoIdeal

Description

Let $\mathcal{P}$ be a collection of cells and $[(1,1),(m,n)]$ be the smallest interval of $\NN^2$ containing $\mathcal{P}$. Then we attach to $\mathcal{P}$ the following polynomial ring $S_{\mathcal{P}}=K[x_a:a\in V(\mathcal{P})$, where $K$ is a field.
Whether it is 1 or by default it returns the ideal computed by polyoIdeal in the ambient ring given by polyoRingDefault. With a value different by 1 it returns the ideal in the ambient ring given by polyoRingConvex.

Here we describe the using of the two options polyoRingDefault and polyoRingConvex.

PolyoRingDefault

This option gives the ideal $I_{\mathcal{P}}$ in the polynomial ring $S_{\mathcal{P}}$ where the monomial order is defined by Term order induced by the following order of the variables: $x_a > x_b$ with $a=(i,j)$ and $b=(k, l)$, if $i > k$, or $i = k$ and $j > l$.

Examples 

i1 : Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};
 
i2 : I = polyoIdeal(Q);

o2 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   ]
                  3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
 
i3 : R=ring I;
 
i4 : describe R

o4 = QQ[x   , x   , x   , x   , x   , x   , x   , x   , Degrees => {8:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}]
         3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1                                                  {Lex => 8          }
                                                                                                        {Position => Up    }
 

i5 : Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};
 
i6 : I = polyoIdeal(Q,RingChoice=>1);

o6 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   ]
                  3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
 
i7 : R=ring I;
 
i8 : describe R

o8 = QQ[x   , x   , x   , x   , x   , x   , x   , x   , Degrees => {8:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}]
         3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1                                                  {Lex => 8          }
                                                                                                        {Position => Up    }
 

i9 : Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};
 
i10 : I = polyoIdeal(Q,RingChoice=>1,TermOrder=> GRevLex);

o10 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   ]
                   3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
 
i11 : R=ring I;
 
i12 : describe R

o12 = QQ[x   , x   , x   , x   , x   , x   , x   , x   , Degrees => {8:1}, Heft => {1}]
          3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1


PolyoRingConvex

A very interesting class of collections of cells which are studied from a combinatorial point of view is given by the weakly connected and convex ones.
Let $\mathcal{P}$ be a collection of cells. We say that $\mathcal{P}$ is weakly connected if for any two cells $C$ and $D$ of $\mathcal{P}$, there exist a sequence of cells of $\mathcal{P}$ as $C = C_1,\dots, C_m = D$ such that $C_i \cap C_{i+1} \neq \emptyset$, for $i = 1,\dots, m − 1$. Observe trivially that every polyomino is a weakly connected collection of cells. We say that a weakly connected collection $\mathcal{P}$ of cells is row convex, if the horizontal cell interval $[A, B]$ is contained in $\mathcal{P}$ for any two cells $A$ and $B$ of $\mathcal{P}$ whose lower left corners are in horizontal position. Similarly one defines column convex. Hence $\mathcal{P}$ is called convex if it is row and column convex.
Assume that the smallest interval containing $\mathcal{P}$ is $[(1,1),(m,n)]$. Consider the edge ring $R = K[s_it_j: (i, j) \in V (\mathcal{P})]$ associated to the bipartite graph $G$ with vertex set $\{s_1,\dots, s_m\} \cup\{t_1,\dots, t_n\}$ to $\mathcal{P}$ such that each vertex $(i, j) \in V (\mathcal{P})$ determines the edge $\{s_i,t_j \}$ in $G$. Let $S=K[x_a:a\in V(\mathcal{P})$ and $\phi : S \rightarrow R$ be the $K$-algebra homomorphism defined by $\phi(x_{ij} ) = s_it_j$, for all $(i, j) \in V (\mathcal{P})$ and set $J_\mathcal{P} = ker(\phi)$. From Theorem 2.1 of [AAQ2012], we know that $I_{\mathcal{P}}=J_{\mathcal{P}}$, if $\mathcal{P}$ is a weakly connected and convex collection of cells. In such a case, from [OH1999] we get that the generators of $I_{\mathcal{P}}$ forms the reduced Groebner basis with respect to a suitable order <, and in particular the initial ideal $\mathrm{in}_<(I_{\mathcal{P}})$ is squarefree and generated in degree two.
Following the proof in [OH1999], this routine implements an algorithm which gives the polynomial ring where the monomial order is <.

If $\mathcal{P}$ is a weakly connected and convex collection of cells (or in particular a convex polyomino), then the function polyoRingConvex returns the polynomial ring attached to $\mathcal{P}$ whose monomial order $<$ is such that $\mathrm{in}_<(I_{\mathcal{P}})$ is squarefree and generated in degree two.

Literature
Examples 

i13 : Q={{{1, 2}, {2, 3}}, {{2, 2}, {3, 3}}, {{1, 3}, {2, 4}}, {{2, 3}, {3, 4}}, {{2, 4}, {3, 5}}, {{3, 2}, {4, 3}}, {{3, 1}, {4, 2}}};
 
i14 : I = polyoIdeal(Q,RingChoice=>2);

o14 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ]
                   2,5   3,5   1,4   2,4   3,4   4,1   3,1   1,2   2,2   4,2   3,2   1,3   2,3   4,3   3,3
 
i15 : R=ring I;
 
i16 : describe R

o16 = QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , Degrees => {15:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, Global => false]
          2,5   3,5   1,4   2,4   3,4   4,1   3,1   1,2   2,2   4,2   3,2   1,3   2,3   4,3   3,3                                                   {RevLex => 15      }
                                                                                                                                                    {Position => Up    }
 


i17 : Q={{{1, 3}, {2, 4}}, {{2, 2}, {3, 3}}, {{2, 3}, {3, 4}}, {{2, 4}, {3, 5}}, {{3, 4}, {4, 5}}, {{3, 3}, {4, 4}}, {{3, 2}, {4, 3}}, {{3, 1}, {4, 2}}, {{3, 5}, {4, 6}}, {{4, 4}, {5, 5}}, {{4, 3}, {5, 4}}, {{5, 4}, {6, 5}}};
 
i18 : I = polyoIdeal(Q,RingChoice=>2);

o18 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ]
                   3,6   4,6   3,1   4,1   2,2   3,2   4,2   1,3   5,3   2,3   3,3   4,3   6,5   5,5   2,5   3,5   4,5   1,4   6,4   5,4   2,4   3,4   4,4
 
i19 : In= monomialIdeal(leadTerm(I))

o19 = monomialIdeal (x   x   , x   x   , x   x   , x   x   , x   x   ,
                      4,6 3,1   4,6 3,2   4,1 3,2   3,2 2,3   4,2 2,3 
      -----------------------------------------------------------------------
      x   x   , x   x   , x   x   , x   x   , x   x   , x   x   , x   x   ,
       4,6 3,3   4,1 3,3   4,2 3,3   2,3 5,5   3,3 5,5   4,3 5,5   3,2 2,5 
      -----------------------------------------------------------------------
      x   x   , x   x   , x   x   , x   x   , x   x   , x   x   , x   x   ,
       4,2 2,5   3,3 2,5   4,3 2,5   4,6 3,5   4,1 3,5   4,2 3,5   4,3 3,5 
      -----------------------------------------------------------------------
      x   x   , x   x   , x   x   , x   x   , x   x   , x   x   , x   x   ,
       5,3 1,4   2,3 1,4   3,3 1,4   4,3 1,4   5,5 6,4   2,5 6,4   3,5 6,4 
      -----------------------------------------------------------------------
      x   x   , x   x   , x   x   , x   x   , x   x   , x   x   , x   x   ,
       4,5 6,4   2,3 5,4   3,3 5,4   4,3 5,4   2,5 5,4   3,5 5,4   4,5 5,4 
      -----------------------------------------------------------------------
      x   x   , x   x   , x   x   , x   x   , x   x   , x   x   , x   x   ,
       3,2 2,4   4,2 2,4   3,3 2,4   4,3 2,4   3,5 2,4   4,5 2,4   4,6 3,4 
      -----------------------------------------------------------------------
      x   x   , x   x   , x   x   , x   x   )
       4,1 3,4   4,2 3,4   4,3 3,4   4,5 3,4

o19 : MonomialIdeal of QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ]
                           3,6   4,6   3,1   4,1   2,2   3,2   4,2   1,3   5,3   2,3   3,3   4,3   6,5   5,5   2,5   3,5   4,5   1,4   6,4   5,4   2,4   3,4   4,4

See also

Functions with optional argument named RingChoice :

For the programmer

The object RingChoice is a symbol.