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A complexity chasm for solving sparse polynomial equations over p -adic fields: extended abstract

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Abstract

The applications of solving systems of polynomial equations are legion: The real case permeates all of non-linear optimization as well as numerous problems in engineering. The p -adic case leads to many classical questions in number theory, and is close to many applications in cryptography, coding theory, and computational number theory. As such, it is important to understand the complexity of solving systems of polynomial equations over local fields. Furthermore, the complexity of solving structured systems --- such as those with a fixed number of monomial terms or invariance with respect to a group action --- arises naturally in many computational geometric applications and is closely related to a deeper understanding of circuit complexity (see, e.g., [8]). Clearly, if we are to fully understand the complexity of solving sparse polynomial systems, then we should at least be able to settle the univariate case, e.g., classify when it is possible to separate and approximate roots in deterministic time polynomial in the input size.

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... Indeed, the underlying root spacing changes dramatically already at 4 terms. While this result goes back to work of Mignotte [20], we point out that in [28] a more general family of polynomials was derived, revealing that the same phenomenon of tightlyspaced roots for tetranomials occurs over all characteristic zero local fields, e.g., the roots of tetranomials in Q p (for p any prime) can be exponentially close as a function of the degree. One may conjecture that the basin of attraction, for Newton's Method applied to a real root of a tetranomial, can also be exponentially small, but so far only the analogous statement over Q p is proved [28,Rem. ...
... While this result goes back to work of Mignotte [20], we point out that in [28] a more general family of polynomials was derived, revealing that the same phenomenon of tightlyspaced roots for tetranomials occurs over all characteristic zero local fields, e.g., the roots of tetranomials in Q p (for p any prime) can be exponentially close as a function of the degree. One may conjecture that the basin of attraction, for Newton's Method applied to a real root of a tetranomial, can also be exponentially small, but so far only the analogous statement over Q p is proved [28,Rem. 4.1]. ...
... Theorem 1.6.[20,29,28] Consider the family of tetranomialsf d (x) := x d − 4 h x 2 + 2 h+2 x − 4 with h ∈ N, h ≥ 3,and d ∈ 4, . . . ...
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... We note that we can be more ambitious and consider also the sparse version as Rojas and Ye [41] in the real world. We note that the results of [1,2,42] impose restrictions for an input that is not random, so the above problems might have a positive solutions. Remark 1.4. ...
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