Changes between Initial Version and Version 1 of Ticket #15422, comment 20
 Timestamp:
 11/29/13 08:02:49 (8 years ago)
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Ticket #15422, comment 20
initial v1 4 4 You can '''never''' say that a padic polynomial has a root. The subvariety of irreducible polynomials is open (in either the Zariski or the padic topology), and any ball will intersect it. So for a given padic polynomial with finite precision, it is either definitely irreducible, or has unknown status. Rather than always raising an `ArithmeticError` instead of factoring, we should make the convention that we will return a factorization to the greatest extent possible among the polynomials within that ball. There is a nice algorithm to determine the precision of the resulting factors. 5 5 6 In particular, the only thing special about polynomials whose discriminant is indistinguishable from zero is that they have the maximum precision loss among reducible polynomials. Among the reducible polynomials in the ball `(1 + O(3^20))*t^2 + (O(3^20))*t + (O(3^20))`, all of them have monic factorizations of the form `((1 + O(3^20))*t + (O(3^10)))*((1 + O(3^20))*t + (O(3^10)))`. For example, (t+3^10 )*(t3^10) would be another possible factorization.6 In particular, the only thing special about polynomials whose discriminant is indistinguishable from zero is that they have the maximum precision loss among reducible polynomials. Among the reducible polynomials in the ball `(1 + O(3^20))*t^2 + (O(3^20))*t + (O(3^20))`, all of them have monic factorizations of the form `((1 + O(3^20))*t + (O(3^10)))*((1 + O(3^20))*t + (O(3^10)))`. For example, (t+3^10^)(t3^10^) would be another possible factorization.