Changes between Initial Version and Version 1 of Ticket #15422, comment 20


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Timestamp:
11/29/13 08:02:49 (8 years ago)
Author:
roed
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  • Ticket #15422, comment 20

    initial v1  
    44You can '''never''' say that a p-adic polynomial has a root.  The subvariety of irreducible polynomials is open (in either the Zariski or the p-adic topology), and any ball will intersect it.  So for a given p-adic 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.
    55
    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)*(t-3^10) would be another possible factorization.
     6In 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^)(t-3^10^) would be another possible factorization.