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The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree

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Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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Abstract

Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be improved by adapting to take advantage of any equational constraints (ECs): equations logically implied by the input. Intuitively, we expect the double exponent in the complexity to decrease by one for each EC. In ISSAC 2015 the present authors proved this for the factor in the complexity bound dependent on the number of polynomials in the input. However, the other term, that dependent on the degree of the input polynomials, remained unchanged.

In the present paper the authors investigate how CAD in the presence of ECs could be further refined using the technology of Gröbner Bases to move towards the intuitive bound for polynomial degree.

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Notes

  1. 1.

    We note that in this quote we made a small correction to the description of the second set of roots (removing a dash from \(y_1\) in the second distinct point). We thank the anonymous referee of the present paper for identifying this correction.

  2. 2.

    It would be possible to economise: if \(x_1x_2^2\mapsto \xi _1^2\), then we could map \(x_1^2x_2^4\) to \(\xi _1^4\) rather than a new \(\xi _2^4\). Since this trick is used purely for the analysis and not in implementation, we ignore such possibilities.

  3. 3.

    As downloaded from www.regularchains.org on 11th March 2016.

  4. 4.

    The On-Line Encyclopedia of Integer Sequences, 2010, Sequence A000124, https://oeis.org/A000124.

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Acknowledgements

This work was originally supported by EPSRC grant: EP/J003247/1 and is now supported by EU H2020-FETOPEN-2016-2017-CSA project \(\mathcal {SC}^2\) (712689). Thanks to the referees for their helpful comments, and Prof. Buchberger for reminding JHD that Gröbner bases were applicable here.

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Authors and Affiliations

  1. Faculty of Engineering, Environment and Computing, School of Computing, Electronics and Maths, Coventry University, Coventry, CV1 5FB, UK

    Matthew England

  2. Department of Computer Science, University of Bath, Bath, BA2 7AY, UK

    James H. Davenport

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  1. Matthew England
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  2. James H. Davenport
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Correspondence to Matthew England .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Appendices

Appendix

A The Iterated Resultants from Section 4

$$\begin{aligned} R_1&:= \mathrm {res}(r_1, r_2, y) = {x}^{16} + 8\,{x}^{15} + ( -8\,{w}^{2}+8\,w+64 ) {x}^{14} + ( -56\,{w}^{2}+56\,w \\ {}&\quad +288 ) {x}^{13} + ( 28\,{w}^{4}-56\,{w}^{3}-332\,{w}^{2}+400\,w+1138 ) {x}^{12} + ( 168\,{w}^{4} \\ {}&\quad -336\,{w}^{3}-1144\,{w}^{2}+1552\,w+2912 ) {x}^{11} + ( -56\,{w}^{6}+168\,{w}^{5}+648\,{w}^{4} \\ {}&\quad -1816\,{w}^{3}-2664\,{w} ^{2}+5328\,w+6336 ) {x}^{10} + ( -280\,{w}^{6}+840\,{w}^{5} \\ {}&\quad +1400\,{w}^{4}-5400\,{w}^{3}-2616\,{w}^{2}+11368\,w+7808 ) {x}^{9} + ( 70\,{w}^{8} \\ {}&\quad -280\,{w}^{7}-500\,{w}^{6}+3080\,{w}^{5}-270\,{w}^{4}-11576\,{w}^{3}+4860\,{w}^{2} \\ {}&\quad +20816\,w+7381 ) {x}^{8} + ( 280\,{w}^{8}-1120\,{w}^{7}+80\,{w}^{6}+6080\,{w}^{5}-8480\,{w}^{4} \\ {}&\quad -11792\,{w}^{3}+22840\,{w}^{2}+20192\,w+920 ) {x}^{7} + ( -56\,{w}^{10}+280\,{w}^{9} \\ {}&\quad -80\,{w}^{8}-2160\,{w}^{7}+4960\,{w}^{6}+3200\,{w}^{5}-22608\,{w}^{4}+2584\,{w}^{3} \\ {}&\quad +40840\,{w}^{2}+16040\,w+2024 ) {x}^{6} + ( -168\,{w}^{10}+840\,{w}^{9}-1520\,{w}^{8} \\ {}&\quad -1360\,{w}^{7}+12016\,{w}^{6}-11296\,{w}^{5}-23368\,{w}^{4}+30136\,{w}^{3}+22032\,{w}^{2} \\ {}&\quad +624\,w+736 ) {x}^{5} + ( 28\,{w}^{12}-168\,{w}^{11}+396\,{w}^{10}+160\,{w}^{9}-3690\,{w}^{8} \\&\quad +6576\,{w}^{7}+4520\,{w}^{6}-24712\,{w}^{5}+13154\,{w}^{4}+37456\,{w}^{3}+1464\,{w}^{2} \\&\quad -1568\,w+5968 ) {x}^{4} + ( 56\,{w}^{12}-336\,{w}^{11}+1192\,{w}^{10}-1680\,{w}^{9} \\&\quad -2688\,{w}^{8}+12496\,{w}^{7}-13464\,{w}^{6}-16912\,{w}^{5}+37240\,{w}^{4}+13472\,{w}^{3} \\ {}&\quad -16384\,{w}^{2}+1984\,w+3072 ) {x}^{3} + ( -8\,{w}^{14}+56\,{w}^{13}-248\,{w}^{12}+520\,{w}^{11} \\&\quad +72\,{w}^{10}-3088\,{w}^{9}+7664\,{w}^{8}-2040\,{w}^{7}-16176\,{w}^{6}+20424\,{w}^{5} \\ {}&\quad +20056\,{w}^{4}-15360\,{w}^{3}-8544\,{w}^{2}+4608\,w+2304 ) {x}^{2} + ( -8\,{w}^{14} \\ {}&\quad +56\,{w}^{13}-296\,{w}^{12}+808\,{w}^{11}-1144\,{w}^{10}-776\,{w}^{9}+6184\,{w}^{8}-7048\,{w}^{7} \\&\quad -6944\,{w}^{6}+19696\,{w}^{5}+3872\,{w}^{4}-16832\,{w}^{3}-1152\,{w}^{2}+4608\,w ) x + {w}^{16} \\&\quad -8\,{w}^{15}+52\,{w}^{14}-184\,{w}^{13}+454\,{w}^{12}-440\,{w}^{11}-772\,{w}^{10} +3352\,{w}^{9} \\ {}&\quad -2447\,{w}^{8}-4288\,{w}^{7}+8200\,{w}^{6}+2080\,{w}^{5}-7664\,{w}^{4} -384\,{w}^{3} + 2304\,{w}^{2}\\ \end{aligned}$$
$$\begin{aligned} R_2&:= \mathrm {res}(r_1, r_3, y) = {x}^{16} + 8\,{x}^{15} + ( -8\,{w}^{2}+8\,w+28 ) {x}^{14} + ( -56\,{w}^{2}+56\,w \\&\quad +48 ) {x}^{13} + ( 28\,{w}^{4}-56\,{w}^{3}-116\,{w}^{2}+160\,w-2 ) {x}^{12} + ( 168\,{w}^{4} \\ {}&\quad -336\,{w}^{3}+80\,{w}^{2}+184\,w-256 ) {x}^{11} + ( -56\,{w}^{6}+168\,{w}^{5}+108\,{w}^{4} \\&\quad -592\,{w}^{3}+852\,{w}^{2}-240\,w-12 ) {x}^{10} + ( -280\,{w}^{6}+840\,{w}^{5}-1120\,{w}^{4} \\&\quad +360\,{w}^{3}+1872\,{w}^{2}-1448\,w+2000 ) {x}^{9} + ( 70\,{w}^{8}-280\,{w}^{7}+220\,{w}^{6} \\&\quad +560\,{w}^{5}-2742\,{w}^{4}+3232\,{w}^{3}-1428\,{w}^{2}+224\,w+4537 ) {x}^{8} + ( 280\,{w}^{8} \\&\quad -1120\,{w}^{7}+2720\,{w}^{6}-3280\,{w}^{5}-1280\,{w}^{4}+6016\,{w}^{3}-11696\,{w}^{2}+7496\,w \\&\quad +2552 ) {x}^{7} + ( -56\,{w}^{10}+280\,{w}^{9}-620\,{w}^{8}+480\,{w}^{7}+2488\,{w}^{6}-6880\,{w}^{5} \\&\quad +9384\,{w}^{4}-5744\,{w}^{3}-9404\,{w}^{2}+12008\,w-4120 ) {x}^{6} + ( -168\,{w}^{10} \\ {}&\quad +840\,{w}^{9} -2960\,{w}^{8}+5840\,{w}^{7}-4832\,{w}^{6}-3088\,{w}^{5}+21104\,{w}^{4} \\ {}&\quad -27128\,{w}^{3} +12552\,{w}^{2}+3888\,w-5888 ) {x}^{5} + ( 28\,{w}^{12}-168\,{w}^{11}+612\,{w}^{10} \\ {}&\quad -1280\,{w}^{9} +498\,{w}^{8}+3648\,{w}^{7}-12424\,{w}^{6}+17360\,{w}^{5}-4546\,{w}^{4} \\&\quad -13928\,{w}^{3} + 19032\,{w}^{2}-9344\,w-176 ) {x}^{4} + ( 56w^{12} - 336w^{11} + 1552w^{10} \\ {}&\quad - 4200\,{w}^{9} + 7296\,{w}^{8} - 6080\,{w}^{7} - 7440\,w^{6}+25880\,{w}^{5}-31352\,{w}^{4} \\ {}&\quad +13472\,{w}^{3} +1856\,{w}^{2}-10304\,w+1536 )x^3 + ( -8\,{w}^{14}+56\,{w}^{13}-284\,{w}^{12} \\ {}&\quad +880\,{w}^{11} -1740\,{w}^{10}+1616\,{w}^{9}+2468\,{w}^{8}-10704\,{w}^{7}+15828\,{w}^{6} \\ {}&\quad -8040\,{w}^{5} -1064\,{w}^{4}+9792\,{w}^{3}-3168\,{w}^{2}+2304 ) {x}^{2} + ( -8\,{w}^{14}+56\,{w}^{13} \\ {}&\quad -320\,{w}^{12} +1096\,{w}^{11}-2800\,{w}^{10}+4600\,{w}^{9}-3968\,{w}^{8}-2152\,{w}^{7} \\ {}&\quad +9592\,{w}^{6} -10832\,{w}^{5} +5312\,{w}^{4}+4672\,{w}^{3}-5760\,{w}^{2}+4608\,w ) x + {w}^{16} \\ {}&\quad -8\,{w}^{15} +52\,{w}^{14}-208\,{w}^{13} +646\,{w}^{12}-1376\,{w}^{11}+2012\,{w}^{10}-1136\,{w}^{9} \\ {}&\quad -1295\,{w}^{8}+4328\,{w}^{7}-3992\,{w}^{6} +2368\,{w}^{5}+2320\,{w}^{4}-1920\,{w}^{3}+2304\,{w}^{2} \\ \end{aligned}$$
$$\begin{aligned} R_3&:= \mathrm {res}(r_3, r_3, y) = {x}^{16} + 8\,{x}^{15} + ( -8\,{w}^{2}+8\,w+44 ) {x}^{14} + ( -56\,{w}^{2}+56\,w \\&\quad +160 ) {x}^{13} + ( 28\,{w}^{4}-56\,{w}^{3}-228\,{w}^{2}+272\,w+430 ) {x}^{12} + ( 168\,{w}^{4}\\&\quad -336\,{w}^{3}-592\,{w}^{2}+856\,w+816 ) {x}^{11} + ( -56\,{w}^{6}+168\,{w}^{5}+444\,{w}^{4} \\&\quad -1264\,{w}^{3}-812\,{w}^{2}+1952\,w+1092 ) {x}^{10} + ( -280\,{w}^{6}+840\,{w}^{5}+560\,{w}^{4} \\&\quad -3000\,{w}^{3}+32\,{w}^{2}+3032\,w+736 ) {x}^{9} + ( 70\,{w}^{8}-280\,{w}^{7}-340\,{w}^{6} \\&\quad +2240\,{w}^{5}-902\,{w}^{4}-4208\,{w}^{3}+2716\,{w}^{2}+3120\,w-183 ) {x}^{8} + ( 280\,{w}^{8} \\&\quad -1120\,{w}^{7}+480\,{w}^{6}+3440\,{w}^{5}-4640\,{w}^{4}-2304\,{w}^{3}+5840\,{w}^{2}+1128\,w\\ {}&\quad -1144 ) {x}^{7} + ( -56\,{w}^{10}+280\,{w}^{9}-60\,{w}^{8}-1760\,{w}^{7}+3128\,{w}^{6}+960\,{w}^{5}\\ {}&\quad -7352\,{w}^{4}+3216\,{w}^{3}+5860\,{w}^{2}-1320\,w-824 ) {x}^{6} + ( -168\,{w}^{10}+840\,{w}^{9}\\ {}&\quad -1280\,{w}^{8}-880\,{w}^{7}+5568\,{w}^{6}-5008\,{w}^{5}-4464\,{w}^{4}+7848\,{w}^{3}+984\,{w}^{2}\\ {}&\quad -2576\,w-64 ) {x}^{5} + ( 28\,{w}^{12}-168\,{w}^{11}+276\,{w}^{10}+400\,{w}^{9}-2302\,{w}^{8}\\ {}&\quad +2848\,{w}^{7}+1880\,{w}^{6}-7440\,{w}^{5}+3582\,{w}^{4}+5704\,{w}^{3}-3208\,{w}^{2}-1216\,w\\ {}&\quad +720 ) {x}^{4} + ( 56\,{w}^{12}-336\,{w}^{11}+880\,{w}^{10}-840\,{w}^{9}-1424\,{w}^{8}+4800\,{w}^{7}\\ {}&\quad -3856\,{w}^{6}-3464\,{w}^{5}+6968\,{w}^{4}+32\,{w}^{3}-3392\,{w}^{2}+448\,w+512 ) {x}^{3} \\ {}&\quad + ( -8\,{w}^{14}+56\,{w}^{13}-172\,{w}^{12}+208\,{w}^{11}+308\,{w}^{10}-1504\,{w}^{9}+1972\,{w}^{8}\\ {}&\quad +432\,{w}^{7}-3788\,{w}^{6}+2920\,{w}^{5}+2552\,{w}^{4}-3136\,{w}^{3}-864\,{w}^{2}+1024\,w\\ {}&\quad +256 ) {x}^{2} + ( -8\,{w}^{14}+56\,{w}^{13}-208\,{w}^{12}+424\,{w}^{11}-352\,{w}^{10}-520\,{w}^{9}\\ {}&\quad +1744\,{w}^{8}-1416\,{w}^{7}-1176\,{w}^{6}+2928\,{w}^{5}-384\,{w}^{4}-1984\,{w}^{3}+384\,{w}^{2}\\ {}&\quad +512\,w ) x + {w}^{16}-8\,{w}^{15}+36\,{w}^{14}-96\,{w}^{13}+150\,{w}^{12}-48\,{w}^{11}-308\,{w}^{10}\\ {}&\quad +672\,{w}^{9}-351\,{w}^{8}-648\,{w}^{7}+1096\,{w}^{6}-880\,{w}^{4}+128\,{w}^{3}+256\,{w}^{2} \end{aligned}$$

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England, M., Davenport, J.H. (2016). The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_12

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