smithNormalForm bug

[pzinn]

found by @mahrud in the discussion of #1273

i1 : debugLevel=101; R=QQ[x]

o2 = R

o2 : PolynomialRing

i3 : smithNormalForm matrix{{x^3+1},{x^2+1}}
-- count = 0
-- lead terms in columns (gb    ) : | 0  |
                                    | x2 |
-- lead terms in rows    (not gb) : | x3 |
                                    | x2 |
-- count = 1
-- lead terms in rows    (gb    ) : {0, 3} | 2 |
-- lead terms in columns (not gb) : {0, 3} | 2 |
-- count = 2
-- lead terms in columns (gb    ) : {0, 3} | 2 |
-- lead terms in rows    (not gb) : {0, 3} | 2 |

o3 = (| 2 |, | x+1       -x2-x+1    |, {3} | 1 |)
      | 0 |  | x2+1      -x3-1      |
      | 0 |  | x3+x2+x+1 -x4-x3-x-1 |

o3 : Sequence

the problem seems to be at the level of the GB computation:

i4 : syz gb (transpose matrix{{x^3+1},{x^2+1}},Syzygies=>true)

o4 = | x2+1  x3+x2+x+1  |
     | -x3-1 -x4-x3-x-1 |

which finds 2 syzygies (?) which confuses the Smith normal form algorithm.

[pzinn]

might as well mention the other issue that I found while investigating this: the first matrix of the output of smithNormalForm depends on the value of ChangeMatrix (even though it probably shouldn't). The reason is pretty simple: when ChangeMatrix is turned on (either for rows or columns), it will add extra zero rows/columns, but not if turned off. The obvious fix is to always add those extra zero rows/columns (which means always calling gb with Syzygies=>true).