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# Randomized polynomial-time root counting in prime power rings

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... The Central Recurrence for Bivariate Point Counting. In this section, we generalize the tools we used for root counting for univariate polynomials in [KRRZ19] to point counting for curves. It is not hard to see that these tools extend naturally to point counting for hypersurfaces of arbitrary dimension. ...
... In particular, by Lemma 2.2, a point ζ is a smooth point off if and only if ζ belongs to just one irreducible componentf r off , the corresponding exponent e r = 1, and ζ is a smooth point off r . Now we are ready to generalize the tools in [KRRZ19] for curves: ...
... In [KRRZ19], we defined a recursive tree structure for root counting for univariate polynomial in Z/p k Z. We define similarly a recursive tree for f (x) = g(x 1 ) + h(x 2 ) that will enable our complexity analysis. ...
Preprint
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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.
... We'll see in Sections 5 and 6 how our speed-up depends on p-adic Diophantine approximation [58,59]. Another key new ingredient in proving Theorem 1.1 is an efficient encoding of roots in Z/(p k ) from [24,36], with an important precursor in [10]. ...
... Definition 2.9. [36] For any f ∈ Z[x] letf denote the mod p reduction of f . Assumef is not identically 0. ...
... Definition 2.11. [36] The set T p,k (f ) naturally admits the structure of a labelled, rooted, directed tree as follows 4 (i) We set f 0,0 := f , k 0,0 := k, and let (f 0,0 , k 0,0 ) be the label of the root node of T p,k (f ). ...
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For any fixed field $K\!\in\!\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5, \ldots\}$, we prove that all polynomials $f\!\in\!\mathbb{Z}[x]$ with exactly $3$ (resp. $2$) monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ within deterministic time $\log^{7+o(1)}(dH)$ (resp. $\log^{2+o(1)}(dH)$) in the classical Turing model: Our underlying algorithm correctly counts the number of roots of $f$ in $K$, and for each such root generates an approximation in $\mathbb{Q}$ with logarithmic height $O(\log^3(dH))$ that converges at a rate of $O\!\left((1/p)^{2^i}\right)$ after $i$ steps of Newton iteration. We also prove significant speed-ups in certain settings, a minimal spacing bound of $p^{-O(p\log^2_p(dH)\log d)}$ for distinct roots in $\mathbb{C}_p$, and even stronger repulsion when there are nonzero degenerate roots in $\mathbb{C}_p$: $p$-adic distance $p^{-O(\log_p(dH))}$. On the other hand, we prove that there is an explicit family of tetranomials with distinct nonzero roots in $\mathbb{Z}_p$ indistinguishable in their first $\Omega(d\log_p H)$ most significant base-$p$ digits.
... So Theorem 1.1 presents a new speed-up, and extends earlier work in [23] where it was shown that detecting roots in Q p for univariate trinomials can be done in NP for fixed p. We'll see in Section 3 how our speed-up depends on applying Yu's Theorem on linear forms in p-adic logarithms [30]. The key new ingredient in proving Theorem 1.1 is an efficient encoding of roots in Z/ p k from [16] (with an important precursor in [4] ...
... Definition 2.8. [16] Let us identify the elements of T p,k (f ) with nodes of a labelled rooted directed tree T p,k (f ) defined inductively as follows 3 : ...
... We call each f i,ζ a nodal polynomial of T p,k (f ). It is in fact possible to easily read off the roots of f in Z/ p k from T p,k (f ) [16]. We will instead use T p,k (f ), with k chosen via our root separation bounds, to efficiently count the roots of f in Z p (and thus in Q p via rescaling). 1 , k). ...
Preprint
We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$, we prove that any polynomial $f\in\mathbb{Z}[x_1]$ with exactly $3$ monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ in deterministic time $\log^{O(1)}(dH)$ in the classical Turing model. (The best previous algorithms were of complexity exponential in $\log d$, even for just counting roots in $\mathbb{Q}_p$.) In particular, our algorithm generates approximations in $\mathbb{Q}$ with bit-length $\log^{O(1)}(dH)$ to all the roots of $f$ in $K$, and these approximations converge quadratically under Newton iteration. On the other hand, we give a unified family of {\em tetra}nomials requiring $\Omega(d\log H)$ bits to distinguish the base-$b$ expansions of their roots in $K$.
... We are unaware of any earlier algorithm achieving this complexity bound, even if randomness is allowed. (A few weeks after our work here was presented at ANTS XIII, an improved complexity bound was obtained in the preprint [20].) It is worth noting that further speed-ups in terms of sparsity (e.g., polynomials with a fixed number of monomial terms) may be difficult to derive: merely deciding the existence of roots in ‫ކ‬ p or ‫ޑ‬ p is already NP-hard (under BPP-reductions) with respect to the sparse encoding [1; 7]. ...
... This volume is the proceedings of the thirteenth ANTS meeting, held July [16][17][18][19][20]2018, at the University of Wisconsin-Madison. It includes revised and edited versions of 28 refereed papers presented at the conference. ...
Article
To whom it may concern: You can freely download all my published papers from my papers page at Texas A&M University: www.math.tamu.edu/~rojas/list2.html
... Breiding [4] made an attempt to generalize homotopy continuation methods, but the metric/topological properties of the p-adics made such an attempt fail. In contrast, subdivision methods are commonplace in the p-adic world [20,36,37] and also in the related world of prime power rings [12,31]. Nevertheless, none of these algorithms seems to use Strassman's theorem as the guiding rule of the subdivision, as Strassman does. ...
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.
... The best known deterministic algorithm has a time complexity exponential in k. We point to [9] and the references therein for more on this. A Las Vegas randomized algorithm for computing the number of roots in the ring Z/p k Z is also given over there which takes time less than some polynomial in terms of k. ...
Preprint
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Nora and Wanda are two players who choose coefficients of a degree $d$ polynomial from some fixed unital commutative ring $R$. Wanda is declared the winner if the polynomial has a root in the ring of fractions of $R$ and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington and Zbarsky to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors.
... There is a long history and very extensive literature dedicated to algorithms on polynomials in finite fields, see, for example [12]. More recently, there was also increasing interest to algorithms for polynomials over residue rings, especially in residue rings modulo prime powers, see [5,9,10,13,17,19,31] and references therein. Here we continue this directions and consider the noisy polynomial interpolation problem modulo prime powers which is analogue to the same problem in finite fields [28,30], which in turn is an extension of the hidden number problem of Boneh and Venkatesan [2,3]. ...
Preprint
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We consider the {\it noisy polynomial interpolation problem\/} of recovering an unknown $s$-sparse polynomial $f(X)$ over the ring $\mathbb Z_{p^k}$ of residues modulo $p^k$, where $p$ is a small prime and $k$ is a large integer parameter, from approximate values of the residues of $f(t) \in \mathbb Z_{p^k}$. Similar results are known for residues modulo a large prime $p$, however the case of prime power modulus $p^k$, with small $p$ and large $k$, is new and requires different techniques. We give a deterministic polynomials time algorithm, which for almost given more than a half bits of $f(t)$ for sufficiently many randomly chosen points $t \in \mathbb Z_{p^k}^*$, recovers $f(X)$.
... [22] improved the time complexity of [3]. Very recently, [19] also found a randomized poly-time algorithm which counts all the roots of f mod p k . ...
Conference Paper
Polynomial factoring has famous practical algorithms over fields-- finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x^2+p \bmod p^2 is irreducible, but x^2+px \bmod p^2 has exponentially many factors! We present the first randomized poly(\deg f, łog p) time algorithm to factor a given univariate integral f(x) modulo p^k, for a prime p and k łeq 4. Thus, we solve the open question of factoring modulo p^3 posed in (Sircana, ISSAC'17). Our method reduces the general problem of factoring f(x) mod p^k to that of \em root finding in a related polynomial E(y) \bmodłangle p^k, \varphi(x)^\ell \rangle for some irreducible \varphi \bmod p. We can efficiently solve the latter for kłe4, by incrementally transforming E(y). Moreover, we discover an efficient refinement of Hensel lifting to lift factors of f(x) \bmod p to those \bmod\ p^4 (if possible). This was previously unknown, as the case of repeated factors of f(x) \bmod p forbids classical Hensel lifting.
... Root counting has interesting applications in arithmetic algebraic-geometry, eg. to compute Igusa zeta function of a univariate integral polynomial [ZG03,DH01]. Partial derandomization of root counting algorithm has been obtained by [CGRW18,KRRZ18] last year; however, a deterministic poly-time algorithm is still unknown. ...
Preprint
Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod p^2$ is irreducible, but $x^2+px \bmod p^2$ has exponentially many factors! We present the first randomized poly(deg $f, \log p$) time algorithm to factor a given univariate integral $f(x)$ modulo $p^k$, for a prime $p$ and $k \leq 4$. Thus, we solve the open question of factoring modulo $p^3$ posed in (Sircana, ISSAC'17). Our method reduces the general problem of factoring $f(x) \bmod p^k$ to that of {\em root finding} in a related polynomial $E(y) \bmod\langle p^k, \varphi(x)^\ell \rangle$ for some irreducible $\varphi \bmod p$. We could efficiently solve the latter for $k\le4$, by incrementally transforming $E(y)$. Moreover, we discover an efficient and strong generalization of Hensel lifting to lift factors of $f(x) \bmod p$ to those $\bmod\ p^4$ (if possible). This was previously unknown, as the case of repeated factors of $f(x) \bmod p$ forbids classical Hensel lifting.
Article
Text For any fixed field K∈{Q2,Q3,Q5,…}, we prove that all univariate polynomials f with exactly 3 (resp. 2) monomial terms, degree d, and all coefficients in {±1,…,±H}, can be solved over K within deterministic time log4+o(1)⁡(dH)log3⁡d (resp. log2+o(1)⁡(dH)) in the classical Turing model: Our underlying algorithm correctly counts the number of roots of f in K, and for each such root generates an approximation in Q with logarithmic height O(log2⁡(dH)log⁡d) that converges at a rate of O((1/p)2i) after i steps of Newton iteration. We also prove significant speed-ups in certain settings, a minimal spacing bound of p−O(plogp2⁡(dH)log⁡d) for distinct roots in Cp, and even stronger root repulsion when there are nonzero degenerate roots in Cp: p-adic distance p−O(logp⁡(dH)). On the other hand, we prove that there is an explicit family of tetranomials with distinct nonzero roots in Zp indistinguishable in their first Ω(dlogp⁡H) most significant base-p digits. So speed-ups for t-nomials with t≥4 will require evasion or amortization of such worst-case instances. Video For a video summary of this paper, please visit https://youtu.be/npfdxLk04MY.
Article
Nora and Wanda are two players who choose coefficients of a degree-d polynomial from some fixed unital commutative ring $R$. Wanda is declared the winner if the polynomial has a root in the ring of fractions of $R$ and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington, and Zbarsky (2018) to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors. Our techniques there lead us to an adversarial approach to constructing rational polynomials of any prescribed degree (equal to 3 or greater than 8) with no roots in the maximal abelian extension of \Bbb{Q}.
Article
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.
Article
We consider the noisy polynomial interpolation problem of recovering an unknown s-sparse polynomial f(X) over the ring Zpk of residues modulo pk, where p is a small prime and k is a large integer parameter, from approximate values of the residues of f(t)∈Zpk. Similar results are known for residues modulo a large prime p, however the case of prime power modulus pk, with small p and large k, is new and requires different techniques. We give a deterministic polynomial time algorithm, which for almost given more than a half bits of f(t) for sufficiently many randomly chosen points t∈Zpk∗, recovers f(X).
Article
Polynomial factoring has famous practical algorithms over fields– finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x2+pmodp2 is irreducible, but x2+pxmodp2 has exponentially many factors in the input size (which here is logarithmic in p)! We present the first randomized poly(deg⁡f,log⁡p) time algorithm to factor a given univariate integral polynomial f modulo pk, for a prime p and k≤4.¹ Thus, we solve the open question of factoring modulo p3 posed in (Sircana, ISSAC'17). Our method reduces the general problem of factoring fmodpk to that of root finding of a related polynomial E(y)mod〈pk,φ(x)ℓ〉 for some irreducible φmodp. We can efficiently solve the latter for k≤4, by incrementally transforming E. Moreover, we discover an efficient refinement of Hensel lifting to lift factors of fmodp to those modp4 (if possible). This was previously unknown, as the case of repeated factors of fmodp forbids classical Hensel lifting.
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