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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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Consider a univariate polynomial f in Z[x] with degree d, exactly t monomial terms, and coefficients in {-H,...,H}. Solving f over the reals, R, in polynomial-time can be defined as counting the exact number of real roots of f and then finding (for each such root z) an approximation w of logarithmic height (log(dH))^{O(1)} such that the Newton iterates of w have error decaying at a rate of O((1/2)^{2^i}). Solving efficiently in this sense, using (log(dH))^{O(1)} deterministic bit operations, is arguably the most honest formulation of solving a polynomial equation over R in time polynomial in the input size. Unfortunately, deterministic algorithms this fast are known only for t=2, unknown for t=3, and provably impossible for t=4. (One can of course resort to older techniques with complexity (d\log H)^{O(1)} for t>=4.) We give evidence that polynomial-time real-solving in the strong sense above is possible for t=3: We give a polynomial-time algorithm employing A-hypergeometric series that works for all but a fraction of 1/Omega(log(dH)) of the input f. We also show an equivalence between fast trinomial solving and sign evaluation at rational points of small height. As a consequence, we show that for "most" trinomials f, we can compute the sign of f at a rational point r in time polynomial in log(dH) and the logarithmic height of r. (This was known only for binomials before.) We also mention a related family of polynomial systems that should admit a similar speed-up for solving.
... 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. ...
Preprint
Solving polynomials is a fundamental computational problem in mathematics. In the real setting, we can use Descartes' rule of signs to efficiently isolate the real roots of a square-free real polynomial. In this paper, we translate this method into the $p$-adic worlds. We show how the $p$-adic analog of Descartes' rule of signs, Strassman's theorem, leads to an algorithm to isolate the roots of a square-free $p$-adic polynomial. Moreover, we show that this algorithm runs in $\mathcal{O}(d^2\log^3d)$-time for a random $p$-adic polynomial of degree $d$. To perform this analysis, we introduce the condition-based complexity framework from real/complex numerical algebraic geometry into $p$-adic numerical algebraic geometry.
... 5.3]). So a long term goal of this work is to improve the complexity of finding the p-adic rational points on curves and surfaces, generalizing recent p-adic speed-ups in the univariate case [RZ20]. ...
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