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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, . . . ...

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. ...

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]. ...

Let $k,p\in \mathbb{N}$ with $p$ prime and let $f\in\mathbb{Z}[x_1,x_2]$ be a bivariate polynomial with degree $d$ and all coefficients of absolute value at most $p^k$. Suppose also that $f$ is variable separated, i.e., $f=g_1+g_2$ for $g_i\in\mathbb{Z}[x_i]$. We give the first algorithm, with complexity sub-linear in $p$, to count the number of roots of $f$ over $\mathbb{Z}$ mod $p^k$ for arbitrary $k$: Our Las Vegas randomized algorithm works in time $(dk\log p)^{O(1)}\sqrt{p}$, and admits a quantum version for smooth curves working in time $(d\log p)^{O(1)}k$. Save for some subtleties concerning non-isolated singularities, our techniques generalize to counting roots of polynomials in $\mathbb{Z}[x_1,\ldots,x_n]$ over $\mathbb{Z}$ mod $p^k$. Our techniques are a first step toward efficient point counting for varieties over Galois rings (which is relevant to error correcting codes over higher-dimensional varieties), and also imply new speed-ups for computing Igusa zeta functions of curves. The latter zeta functions are fundamental in arithmetic geometry.

Given any complex Laurent polynomial f, Amoeba(f) is the image of the complex
zero set of f under the coordinate-wise log absolute value map. We give an
efficiently constructible polyhedral approximation, ArchTrop(f), of Amoeba(f),
and derive explicit upper and lower bounds, solely as a function of the
sparsity of f, for the Hausdorff distance between these two sets. We thus
obtain an Archimedean analogue of Kapranov's Non-Archimedean Amoeba Theorem and
a higher-dimensional extension of earlier estimates of Mikhalkin and Ostrowski.
We then show that deciding whether a given point is in ArchTrop(f) is doable in
polynomial-time, for any fixed dimension, unlike the corresponding question for
Amoeba(f), which is np-hard already in one variable.

Let $k$ be a locally compact complete field with respect to a discrete
valuation $v$. Let $\oo$ be the valuation ring, $\m$ the maximal ideal and
$F(x)\in\oo[x]$ a monic separable polynomial of degree $n$. Let
$\delta=v(\dsc(F))$. The Montes algorithm computes an OM factorization of $F$.
The single-factor lifting algorithm derives from this data a factorization of
$F \md{\m^\nu}$, for a prescribed precision $\nu$. In this paper we find a new
estimate for the complexity of the Montes algorithm, leading to an estimation
of $O(n^{2+\epsilon}+n^{1+\epsilon}\delta^{2+\epsilon}+n^2\nu^{1+\epsilon})$
word operations for the complexity of the computation of a factorization of $F
\md{\m^\nu}$, assuming that the residue field of $k$ is small.

Kurt Hensel (1861-1941) discovered the p-adic numbers around the turn of the century. These exotic numbers (or so they appeared at first) are now well-established in the mathematical world and used more and more by physicists as well. This book offers a self-contained presentation of basic p-adic analysis. The author is especially interested in the analytical topics in this field. Some of the features which are not treated in other introductory p-adic analysis texts are topological models of p-adic spaces inside Euclidean space, a construction of spherically complete fields, a p-adic mean value theorem and some consequences, a special case of Hazewinkel's functional equation lemma, a remainder formula for the Mahler expansion, and most importantly a treatment of analytic elements.

We give a separation bound for the complex roots of a trinomial $f \in \mathbb{Z}[X]$. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of $f$; in particular, it is polynomial in $\log (\deg f)$. It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of $f$ rather than the number of monomials) give separation bounds that are exponentially worse.As an algorithmic application, we show that the number of real roots of a trinomial $f$ can be computed in time polynomial in the size of the sparse encoding of~$f$. The same problem is open for 4-nomials.

Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.

We give an efficient algorithm for factoring polynomials over finite algebraic extensions of the ρ-adic numbers. This algorithm uses ideas of Chistov’s random polynomial-time algorithm, and is suitable for practical implementation.

This article is devoted to algorithms for computing all the roots of a univariate polynomial with coefficients in a complete commutative Noetherian unramified regular local domain, which are given to a fixed common finite precision. We study the cost of our algorithms, discuss their practical performances, and apply our results to the Guruswami and Sudan list decoding algorithm over Galois rings.

We prove a lower bound for the distance between two roots of a polynomial with complex coefficients. Such estimates are used to separate the roots of a polynomial.

Let $f(x)$ be a separable polynomial over a local field. Montes algorithm
computes certain approximations to the different irreducible factors of $f(x)$,
with strong arithmetic properties. In this paper we develop an algorithm to
improve any one of these approximations, till a prescribed precision is
attained. The most natural application of this "single-factor lifting" routine
is to combine it with Montes algorithm to provide a fast polynomial
factorization algorithm. Moreover, the single-factor lifting algorithm may be
applied as well to accelerate the computational resolution of several global
arithmetic problems in which the improvement of an approximation to a single
local irreducible factor of a polynomial is required.

We present algorithms revealing new families of polynomials allowing sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we show that the case of honest n-variate (n+1)-nomials is doable in NP and, for p exceeding the constant term and the Newton polytope volume, in constant time. Furthermore, using the theory of linear forms in p-adic logarithms, we prove that the case of trinomials in one variable can be done in NP. The best previous complexity bounds for these problems were EXPTIME or worse. Finally, we prove that detecting p-adic rational roots for sparse polynomials in one variable is NP-hard with respect to randomized reductions. Our proof makes use of an efficient construction of primes in certain arithmetic progressions. The smallest n where detecting p-adic rational roots for n-variate sparse polynomials is NP-hard appears to have been unknown. Comment: 28 pages, 3 figures, submitted for publication

This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z=rZ for r 2 Z or F q [y]=rF q [y] for r 2 F q [y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discriminant of f , our (probabilistic) algorithm produces a description of all (possibly exponentially many) factorizations into irreducible factors in polynomial time. If r divides the discriminant, we only know how to factor by exhaustive search, in exponential time.

This article has appeared in Math. Res. Lett. 5 (1998), 273--279, c #MRL 1998. 1 2 BJORN POONEN uniformly the number of factors of given degree over number fields. In [Le2] he shows that if f is represented sparsely, then these factors can be found in polynomial time. Remark. We cannot count multiplicities in either of our theorems and hope to obtain a bound depending only on k and K (and d, for Theorem 2), because of examples like f(x) = (1+x) q m with m ##. Requiring that f not be a p-th power would not eliminate the problem, because one could also take f(x) = (1 + x) q m +1 . 2. Proof of Theorem 1 By a disk in a valued field K, we mean either an "open disk" D(x 0 , g) := {x # K : v(x - x 0 ) > g}, or a "closed disk" D(x 0 , g) := {x # K : v(x - x 0 ) # g} where x 0 # K and g # G. Let # 1 , # 2 , . . . , # t be the non-vertical segments of the Newton polygon of f . Let -g j # G# Q be the slope of # j . If e 1 , e 2 , . . . , e r are the exponents of the monomials in f corresponding to points on a given # j , define N j as the largest integer for which the images of (1 + x) e 1 , (1 +x) e 2 , . . . , (1 +x) er in F p [x]/(x N j ) are linearly dependent over F p . We say that the # j are in a proper order if N 1 # N 2 # # N t . This particular ordering is crucial to the proof, but it is hard to motivate its definition. It was discovered by analyzing proofs of many special cases of Theorem 1. For instance, if the Newton polygon of f has k non-vertical segments (each associated with exactly two exponents), then the segments are being ordered according to the p-adic absolute values of their horizontal lengths. Lemma 3. Let L be a field of characteristic p > 0 with a valuation v : L # # G. Suppose f(x) = a 0 x n 0 + a 1...