Defactoring algorithms: Difference between revisions

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We're actually quite close to done! All we need to do is flip the signs on the 2nd row. But wait, you protest! Isn't that multiplying a row by −1, which we specifically forbade? Well, sure, but that just shows we need to clarity what we're concerned about, which is essentially enfactoring. Multiplying by −1 does not change the GCD of the row, where multiplying by −2 or 2 would. Note that because the process for taking the HNF forbids multiplying ''or dividing'', it will never introduce enfactoring where was there was none previously, but it also does not remove enfactoring that is there.

Perhaps another helpful way of thinking about this is that multiplying the row by -1 does not alter the potential effects this row could have being added or subtracted from other rows. It merely swaps addition and subtraction. Whereas multiplying the row by any integer with absolute value greater than 1 ''would'' affect the potential effects this row could have being added or subtracted from other rows: it would limit them.

Perhaps another helpful way of thinking about this is that multiplying the row by −1 does not alter the potential effects this row could have being added or subtracted from other rows. It merely swaps addition and subtraction. Whereas multiplying the row by any integer with absolute value greater than 1 ''would'' affect the potential effects this row could have being added or subtracted from other rows: it would limit them.

So, let's do that sign flip:

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 We're actually quite close to done! All we need to do is flip the signs on the 2nd row. But wait, you protest! Isn't that multiplying a row by &minus;1, which we specifically forbade? Well, sure, but that just shows we need to clarity what we're concerned about, which is essentially enfactoring. Multiplying by &minus;1 does not change the GCD of the row, where multiplying by &minus;2 or 2 would. Note that because the process for taking the HNF forbids multiplying ''or dividing'', it will never introduce enfactoring where was there was none previously, but it also does not remove enfactoring that is there.
-Perhaps another helpful way of thinking about this is that multiplying the row by -1 does not alter the potential effects this row could have being added or subtracted from other rows. It merely swaps addition and subtraction. Whereas multiplying the row by any integer with absolute value greater than 1 ''would'' affect the potential effects this row could have being added or subtracted from other rows: it would limit them.
+Perhaps another helpful way of thinking about this is that multiplying the row by &minus;1 does not alter the potential effects this row could have being added or subtracted from other rows. It merely swaps addition and subtraction. Whereas multiplying the row by any integer with absolute value greater than 1 ''would'' affect the potential effects this row could have being added or subtracted from other rows: it would limit them.
 So, let's do that sign flip: