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Composition Modulo Powers of Polynomials

Joris van der Hoeven, Grégoire Lecerf
<span title="">2017</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p5cu7ybzmnd3jpp7pphgogfzpi" style="color: black;">Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC &#39;17</a> </i> &nbsp;
In this paper, we study the more specic case of composition modulo the power of a polynomial. First we extend previously known algorithms for power series composition to this context.  ...  Modular composition is the problem to compose two univariate polynomials modulo a third one.  ...  Using Chinese remaindering, one may reduce composition modulo such h to compositions modulo powers of irreducible polynomials.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3087604.3087634">doi:10.1145/3087604.3087634</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/issac/HoevenL17a.html">dblp:conf/issac/HoevenL17a</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pt4wiyzzvrcunj4wutqn45ckqy">fatcat:pt4wiyzzvrcunj4wutqn45ckqy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190505171909/https://hal.archives-ouvertes.fr/hal-01455722/file/pcomp.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/45/05/4505eeacb2092604fbf13c6e8aa5c309e887ced2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3087604.3087634"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?

A J Guttmann, I Jensen, J-M Maillard, J Pantone
<span title="2016-11-21">2016</span> <i title="IOP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nzcpmvl7nzgojf4q47bpg2qrdu" style="color: black;">Journal of Physics A: Mathematical and Theoretical</a> </i> &nbsp;
We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions.  ...  coefficients might reduce to algebraic functions modulo primes, or powers of primes.  ...  This type of behaviour of differential approximants is characteristic of an irregular singularity.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1751-8113/49/50/504002">doi:10.1088/1751-8113/49/50/504002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ekocc5trrvcablzz7axad7omiq">fatcat:ekocc5trrvcablzz7axad7omiq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170111112948/http://www.ms.unimelb.edu.au/~ij@unimelb/Publications/2016/JPA_49_504002.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/62/4b/624b44fdde97177eaee27e42738711067ad1265a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1751-8113/49/50/504002"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> iop.org </button> </a>

Near Quadratic Matrix Multiplication Modulo Composites [article]

Vince Grolmusz
<span title="2003-02-04">2003</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show how one can use non-prime-power, composite moduli for computing representations of the product of two n× n matrices using only n^2+o(1) multiplications.  ...  The author acknowledges the partial support of an NKFP grant and an ETIK grant and the EU FP 5 grant IST FET IST-2001-32012.  ...  The main result of the present paper is an algorithm with n 2+o(1) multiplications for computing a representation of the matrix product modulo non-prime power composite numbers (e.g., 6).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/cs/0301004v3">arXiv:cs/0301004v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y6ilri6a2ng3fdbwmgwfczexpa">fatcat:y6ilri6a2ng3fdbwmgwfczexpa</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-cs0301004/cs0301004.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c6/f9/c6f97088eacaa0ab9c28bc70df39ef77dd8ef545.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/cs/0301004v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Primality Tests Based on Fermat's Little Theorem [chapter]

Manindra Agrawal
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this survey, we describe three algorithms for testing primality of numbers that use Fermat's Little Theorem.  ...  Since elements of B 0 are polynomials modulo h(x) and degree of h(x) ≤ r − 1, T ≤ p r−1 . The lower bound on T is a little more involved. Consider any two polynomials f (x), g(x) ∈ B of degree < t.  ...  Assume that n is composite but not a prime power. Let p and q be two odd prime divisors of n. Let k be the largest power of p dividing n. Let p − 1 = 2 v · w where w is odd.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11947950_32">doi:10.1007/11947950_32</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hjrwubtxjndareovn42gt4b3yy">fatcat:hjrwubtxjndareovn42gt4b3yy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808134334/https://www.cse.iitk.ac.in/users/manindra/survey/FLTBasedTests.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d6/39/d639c3bce404aacbf4da36e5034a3743926404b5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11947950_32"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Modular composition via factorization

Joris van der Hoeven, Grégoire Lecerf
<span title="">2018</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hhnp3o4qcjfyfd4gpifcdou6ui" style="color: black;">Journal of Complexity</a> </i> &nbsp;
Modular composition is the problem to compute the composition of two univariate polynomials modulo a third one.  ...  In this article, we explore particular cases of moduli over finite fields for which modular composition turns out to be cheaper than in the general case.  ...  We introduce cost functions for modular composition, power projection, and the computation of characteristic polynomials.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jco.2018.05.002">doi:10.1016/j.jco.2018.05.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dcqo5md4arbcrgwxw2ak634l3m">fatcat:dcqo5md4arbcrgwxw2ak634l3m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190504003703/https://hal.archives-ouvertes.fr/hal-01457074/file/ffcomp.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/eb/f8/ebf808046d9b72e6f8db9c260ddacf96e8037290.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jco.2018.05.002"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Modular Representations of Polynomials: Hyperdense Coding and Fast Matrix Multiplication

Vince Grolmusz
<span title="">2008</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
A certain modular representation of multilinear polynomials is considered. The modulo 6 representation of polynomial f is just any polynomial f + 6g.  ...  The 1-a-strong representation of f modulo 6 is polynomial f + 2g + 3h, where no two of g, f and h have common monomials.  ...  If we choose a non-prime-power, composite modulus, say m = 6, then the modulo 6 representation of polynomial f is also a modulo 3 and modulo 2 representation at the same time.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tit.2008.926346">doi:10.1109/tit.2008.926346</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qvvq7tkz7zhgzederxpkkiopae">fatcat:qvvq7tkz7zhgzederxpkkiopae</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810113128/http://bolyai.cs.elte.hu/%7Egrolmusz/papers/hiperdense-info3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/50/1f/501f439df0c59188be9acde2328764c14dd33c89.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tit.2008.926346"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Faster Modular Composition [article]

Vincent Neiger, Bruno Salvy, Éric Schost, Gilles Villard
<span title="2021-10-15">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field.  ...  The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require O(n^1.63) field operations in general, and n^3/2+o(1) field operations in the particular case of power series over a field  ...  For a modulus of the form ℎ( ) ℓ , van der Hoeven and Lecerf showed how composition can be reduced to ℓ compositions modulo ℎ, the computation of an annihilating polynomial modulo ℎ, and a power series  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2110.08354v1">arXiv:2110.08354v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r4abnf5vundljczjbfdfh75axu">fatcat:r4abnf5vundljczjbfdfh75axu</a> </span>
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Page 241 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 22, Issue 2 [page]

<span title="">1921</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_american-mathematical-society-transactions" style="color: black;">American Mathematical Society. Transactions of the American Mathematical Society </a> </i> &nbsp;
We shall call the polynomials of this set “completely reduced polynomials”? modulo m ($5).  ...  general composite modulus.  ... 
<span class="external-identifiers"> </span>
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On set systems with restricted intersections modulo a composite number [chapter]

Vince Grolmusz
<span title="">1997</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Frankl and Wilson asked in 3] whether an analogous upper bound existed for non{prime power, composite moduli.  ...  Here we show a surprising construction of a superpolynomial{sized uniform set-system H satisfying the intersection{property, for every non{prime{power, composite m, negatively settling a related conjecture  ...  The author acknowledges the support of grant OTKA F014919.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bfb0045075">doi:10.1007/bfb0045075</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/poyuciz4o5gqpepottutju2qrq">fatcat:poyuciz4o5gqpepottutju2qrq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190216032853/http://pdfs.semanticscholar.org/00b8/823d92fe84d4a40fcc6e408bfc6db22a416e.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/00/b8/00b8823d92fe84d4a40fcc6e408bfc6db22a416e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bfb0045075"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Fast Polynomial Factorization and Modular Composition

Kiran S. Kedlaya, Christopher Umans
<span title="">2011</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7dys7zoberdktmxyjciuy5bnse" style="color: black;">SIAM journal on computing (Print)</a> </i> &nbsp;
The improvements come from new algorithms for modular composition of degree n univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields.  ...  We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent α implies an algorithm for the  ...  Finally, we thank Madhu Sudan for hosting a visit of the second author to MIT, which launched this collaboration.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/08073408x">doi:10.1137/08073408x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vm32on54svfizhtnvgx6h7zfmm">fatcat:vm32on54svfizhtnvgx6h7zfmm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20100717091757/http://www.cs.caltech.edu/%7Eumans/papers/KU08-final.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/68/36/6836e880ea7a8791e0b29eb9f2fe7b62b9e04fe4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/08073408x"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 1793 of Mathematical Reviews Vol. , Issue 2004c [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
power modulo p.  ...  For n >3 the author derives a similar condition, applicable to a polynomial f all of whose points are singular modulo p and is a PP modulo p’, that guarantees that f is a PP modulo p’, /> 2.  ... 
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Modular Composition Modulo Triangular Sets and Applications

Adrien Poteaux, Éric Schost
<span title="2013-04-18">2013</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t46w5lpc3ngnfmzil33zdjjrp4" style="color: black;">Computational Complexity</a> </i> &nbsp;
We generalize Kedlaya and Umans' modular composition algorithm to the multivariate case.  ...  For the first time, we are able to exhibit running times for these operations that are almost linear, without any overhead exponential in the number of variables.  ...  Modular composition and power projection In this section, we give our first algorithms for multivariate modular composition and power projection; we work modulo a triangular set T of multidegree d ∈ N  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00037-013-0063-y">doi:10.1007/s00037-013-0063-y</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v52lbh5mtbbmdgympfgfjv6hby">fatcat:v52lbh5mtbbmdgympfgfjv6hby</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922003621/http://www.csd.uwo.ca/%7Eeschost/publications/mulmodcomp.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/80/ac/80acf71230a0d603a33c9ac67739a8453bdb631d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00037-013-0063-y"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

A generalization of Miller's primality theorem

Pedro Berrizbeitia, Aurora Olivieri
<span title="2008-05-07">2008</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
For any integer r we show that the notion of ω-prime to base a introduced by Berrizbeitia and Berry, 2000, leads to a primality test for numbers n congruent to 1 modulo r, which runs in polynomial time  ...  Acknowledgments We thank Andrew Shallue for his careful revision of our preprint and his various valuable remarks, which helped to improve the overall quality of the paper.  ...  We also thank the "Decanato de Investigación y Desarrollo de la Universidad Simón Bolívar" for their financial support of our research group GID-24.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9939-08-09303-9">doi:10.1090/s0002-9939-08-09303-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/twu2hgetkfetjjudxsxjzceubi">fatcat:twu2hgetkfetjjudxsxjzceubi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190501043306/https://www.ams.org/journals/proc/2008-136-09/S0002-9939-08-09303-9/S0002-9939-08-09303-9.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a1/f5/a1f5f557cd3d3223f62b79428d4adaee8d7b5b26.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9939-08-09303-9"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Reciprocals of exponential polynomials and permutation enumeration [article]

Ira M. Gessel
<span title="2019-05-17">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
to 0 or 1 modulo 2m.  ...  We show that the reciprocal of a partial sum with 2m terms of the alternating exponential series is the exponential generating function for permutations in which every increasing run has length congruent  ...  A composition is a finite (possibly empty) sequence of positive integers. A composition of n is a composition with sum n.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1807.09290v3">arXiv:1807.09290v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hmgnyvibzrgx7ot2q35p45mziy">fatcat:hmgnyvibzrgx7ot2q35p45mziy</a> </span>
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On the complexity of computing with zero-dimensional triangular sets

Adrien Poteaux, Éric Schost
<span title="">2013</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
We generalize Kedlaya and Umans' modular composition algorithm to the multivariate case.  ...  For the first time, we are able to exhibit running times for these operations that are almost linear, without any overhead exponential in the number of variables.  ...  Modular composition and power projection In this section, we give our first algorithms for multivariate modular composition and power projection; we work modulo a triangular set T of multidegree d ∈ N  ... 
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