Efficient p-adic Cell Decompositions for Univariate Polynomials

Maller, Michael; Whitehead, Jennifer

doi:10.1006/jcom.1999.0520

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…By [11,Lemma 4.1], the complexity of this step is O(n 5 p(log 2 (L) + n 2 ) = C 3 , where L is the precision computed in the previous step. Therefore, we obtain the following complexity estimate Theorem 2.…”

Section: Algorithm 3 Call Ecd(l)mentioning

confidence: 99%

“…The main technical tool is the cell decomposition algorithm of restricted precision L, ECD(L) which we defined in [11,Algorithm 3.2,p. 521].…”

Section: Introductionmentioning

confidence: 99%

“…The cell decompositions ECD(L) are constructed in [11] by working up through the derivatives of k(x). The assumption degree(k(x)) = n < p is used to control the proliferation of cells, leading to a polynomial-time algorithm sufficiently strong to produce accurate root counting and approximation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the complexity of p-adic basic semi-algebraic sets

Maller

Whitehead

2007

Journal of Complexity

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Section: Algorithm 3 Call Ecd(l)mentioning

confidence: 99%

“…The main technical tool is the cell decomposition algorithm of restricted precision L, ECD(L) which we defined in [11,Algorithm 3.2,p. 521].…”

Section: Introductionmentioning

confidence: 99%

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On the complexity of p-adic basic semi-algebraic sets

Maller

Whitehead

2007

Journal of Complexity

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“…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.…”

Section: The Strassman Solvermentioning

confidence: 99%

A $p$-adic Descartes solver: the Strassman solver

Tonelli-Cueto¹

2022

Preprint

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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 padic 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 O(d 2 log 3 d)-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.

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“…Paraphrased in our notation, Erich Kaltofen asked in 2003 whether FEAS Z/pZ (F 1,3 ) admits a (deterministic) algorithm with complexity (log(p) + size(f )) O(1) [Kal03]. 5 The best previous complexity upper bound for FEAS Q primes (Z[x 1 ]×P) relative to the sparse input size appears to have been EXPTIME [MW99].…”

Section: Introductionmentioning

confidence: 99%

Faster p-adic Feasibility for Certain Multivariate Sparse Polynomials

Avendaño¹,

Ibrahim²,

Rojas³

et al. 2010

Preprint

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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 Newton polytope volume and not dividing any coefficient, 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. The last 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.

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Efficient p-adic Cell Decompositions for Univariate Polynomials

Cited by 5 publications

References 7 publications

On the complexity of p-adic basic semi-algebraic sets

On the complexity of p-adic basic semi-algebraic sets

A $p$-adic Descartes solver: the Strassman solver

Faster p-adic Feasibility for Certain Multivariate Sparse Polynomials

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