MR2172144
Orange, Sébastien(F-PARIS6-IP6); Renault, Guenaël(F-PARIS6-IP6); Valibouze, Annick(F-PARIS6-IP6)
Note sur les relations entre les racines d'un polynôme réductible. (French. English, French summary) [Note on the relations between the roots of an irreducible polynomial]
Theor. Inform. Appl. 39 (2005), no. 4, 651--659.
12Fxx

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{A review for this item is in process.}

MR2141285
Valibouze, Annick(F-PARIS6)
Dépendance algébrique des zéros de polynômes et groupes de Galois. (French. English summary) [Algebraic dependence of the zeros of polynomials and Galois groups]
Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48(96) (2005), no. 1, 73--96.
12F10

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{A review for this item is in process.}


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MR2104299 (2005h:13036)
Abdeljaouad-Tej, I.; Orange, S.(F-PARIS6-IP6); Renault, G.(F-PARIS6-IP6); Valibouze, A.(F-PARIS6-IP6)
Computation of the decomposition group of a triangular ideal. (English. English summary)
Appl. Algebra Engrg. Comm. Comput. 15 (2004), no. 3-4, 279--294.
13F20

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Let $R=K[X_1,X_2,\dots,X_n]$ be a polynomial ring over a perfect field $K$. The symmetric group $S_n$ acts naturally on this ring. Let $I$ be an ideal of this ring generated by a triangular set of $n$ polynomials of $R$. The stabilizer ${\rm Dec}(I)$ of $I$ is called the decomposition group.

In this paper, the authors use the backtracking technique of G. Butler \ref[ Fundamental algorithms for permutation groups, Lecture Notes in Comput. Sci., 559, Springer, Berlin, 1991; MR1225579 (94d:68049)] to develop algorithms 2.1 and 3.8, utilizing the MAGMA algebra system software, to study the decomposition group of ${\rm Dec}(I)$.

In Section 3, it is shown that if ${\rm Dec}(I)=S_n$, algorithm 2.1 computes only $n(n+1)$ membership tests in order to determine a generating set composed of $n-1$ transpositions. In Section 4, for a triangular ideal $I$, computation of ${\rm Dec}(I)$ with algorithm 3.8 leads to the following cases: \roster \item A backtrack appears and $I$ is not pure Galois. \item $I$ is pure Galois if and only if ${\rm Card}({\rm Dec}(I))={\rm cardinal}$ of the variety of $I$. \endroster Section 5 is devoted to comparison of various algorithms.

Reviewed by R. K. Markanda
Previous Item

MR1800031 (2002a:12004)
Aubry, Philippe(F-PARIS6-IP6); Valibouze, Annick(F-PARIS6-IP6)
Using Galois ideals for computing relative resolvents. (English. English summary)
Algorithmic methods in Galois theory.
J. Symbolic Comput. 30 (2000), no. 6, 635--651.
12F10 (12Y05)

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Let $f\in k[X]$ be a separable polynomial of degree $n$, where $k$ is a perfect field. Let $\overline k$ be the algebraic closure of $k$ and $\Omega=(\alpha _1 ,\dots, \alpha _n)\in{\overline k}{}^n$, where $\alpha _1 , \dots, \alpha _n$ are the distinct roots of $f$. If $L$ is a subgroup of the symmetric group $\Sigma _n$, the authors define the Galois $(L,\Omega)$-ideal $I_\Omega ^L $ \rm as $I_\Omega ^L =\scr{J}(V)$, where $V=\{\sigma\Omega| \sigma\in L\}$ and $\scr{J}$$(V)=\{f\in k[ x_1,\dots,x_n ]| f(\beta)=0$, for all $\beta\in V\}$, a radical ideal in $k[x_1,\dots,x_n]$. Proposition 5.2 shows that $V$ is an "equiprojectable variety" in ${\overline k}{}^n$\rm (a geometric concept introduced in the paper which means that for all $1\leq i\leq n$, and for all $M\in \pi _i(V)$, card$(\pi _i^{-1}(M) \cap V)$ is finite and does not depend on $M$; $\pi _i :V \to \overline k{}^i $ sends $(a_1 ,\dots, a _n)$ to $(a_1 ,\dots, a _i)$). A zero-dimensional variety $U$ is equiprojectable if and only if $\scr{J}$$(U)$ is an ideal generated by a separable triangular set of polynomials \rm (Theorem 4.5). It follows (Theorem 5.3) that the ideal $I_\Omega ^L $ has a Gröbner basis consisiting of a separable triangular set of polynomials. For an ideal $I$ of this type, an algorithm for computing the characteristic polynomial of the multiplication by a polynomial $\Theta$ inside $k[x_1,\dots,x_n]/I$ is given in Section 6. This algorithm yields a method for computing relative resolvents in Galois theory, illustrated by the case of an irreducible polynomial of degree 8 over $\Bbb{Q}$. A discussion of various implementation and optimization issues is included.

{For the entire collection see MR1800030 (2001f:12001).}

Reviewed by Aurelian Claudiu Volf
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[References]

Note:

This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. Abdeljaouad, I. (1999). Calcul d'invariants primitifs de groupes finis. RAIRO---Inform. Th\'eor. Programm., 33, 59--77. MR1705856 (2000i:13006)
2. Arnaudiès, J., Valibouze, A. (1996). Lagrange resolvents. In Cohen, A. and Roy, M., eds, Special Issue of MEGA '96, J. Pure Appl. Algebra, 117 \& 118, 23--40. MR1457831 (98f:12004)
3. Aubry, P. (1999). Ensembles triangulaires de polynômes et résolution de systèmes algébriques. Ph.D. Thesis, Université Paris 6.
4. Aubry, P., Moreno Maza, M (1999). Triangular sets for solving polynomial systems: a comparative implementation of four methods. J. Symb. Comput., 28, 125--154. MR1709420 (2000g:13017)
5. Becker, T., Weispfenning, V. (1993). Gr\"obner Bases, volume 141 of Graduate Texts in Mathematics. New York, Springer-Verlag. MR1213453 (95e:13018)
6. Buchberger, B. (1965). Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph.D. Thesis, Universität Innsbruck.
7. Butler, G., McKay, J. (1983). The transitive groups of degree up to 11. Commun. Algebra, 11, 863--911. MR0695893 (84f:20005)
8. Casperson, D., McKay, J. (1994). Symmetric functions, $m$-sets, and Galois groups. Math. Comput., 63, 749--757. MR1234424 (95a:12001)
9. Colin, A. (1989). Formal computation of Galois groups with relative resolvents. In Cohen, G., Giusti, M. and Mora, T., eds, Proceedings of AAECC'11, Paris, July 1995, LNCS 948, pp. 169--182. Berlin, Springer-Verlag. MR1448163 (98e:12001)
10. Darmon, H., Ford, D. (1989). Computational verification of $M_{11}$ and $M_{12}$ as Galois groups over $\Bbb{Q}$. Commun. Algebra, 17, 2941--2943. MR1030603 (91b:11146)
11. Eichenlaub, Y. (1996). Problèmes effectifs de théorie de Galois en degrés 8 à 11. Ph.D. Thesis, Université de Bordeaux 1.
12. Faugère, J. (1999). A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra, 139, 61--88. MR1700538 (2000c:13038)
13. Geissler, K., Klüners, J. (2000). Galois group computation for rational polynomials. J. Symb. Comput, 30, 653--674, doi:10.1006/jsco.2000.0377. MR1800032 (2001k:12006)
14. Henrici, P. (1956). Automatic computations with power series. J. Assoc. Comput. Mach., 3, 10--15. MR0074958 (17,673f)
15. Lagrange, J. (1770). Réflexions sur la résolution algébrique des équations. Mémoires de l'Académie de Berlin.
16. Lazard, D. (1991). Systems of algebraic equations (algorithms and complexity). In Eisenbud, D. and Robbiano, L., eds, Cortona Proceedings. Cambridge, Cambridge University Press. MR1253989 (94m:14076)
17. Lazard, D. (1992). Solving zero-dimensional algebraic systems. J. Symb. Comput., 15, 117--132. MR1153638 (93a:68066)
18. Lehobey, F. (1997). Resolvent computations by resultants without extraneous powers. In Küchlin, W., ed., Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 85--92. New York, ACM Press. MR1809974
19. Rennert, N., Valibouze, A. (1999). Calcul de résolvantes avec les modules de Cauchy. Exp. Math., 8, 351--366. MR1737231 (2001e:12002)
20. Soicher, L. (1981). The computations of Galois groups. Ph.D. Thesis, Concordia University, Montreal.
21. Soicher, L., McKay, J. (1985). Computing Galois groups over the rationals. J. Number Theory, 20, 273--281. MR0797178 (87a:12002)
22. Stauduhar, R. (1973). The determination of Galois groups. Math. Comput., 27, 981--996. MR0327712 (48 #6054)
23. Valibouze, A. (1989a). Résolvantes et fonctions symétriques. In Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, pp. 390--399. New York, ACM Press.
24. Valibouze, A. (1989b). Sym, symbolic computation with symmetric polynomials: an extension to macsyma. In Kalthofen, E. and Watt, S. M., eds, Proceedings of Computers and Mathematics (Cambridge, Massachussetts, 1989). New York, Springer-Verlag.
25. Valibouze, A. (1995). Computation of the Galois group of the resolvent factors for the direct and inverse Galois problems. In Cohen, G., Giusti, M. and Mora, T., eds, Proceedings of AAECC'11, Paris, July 1995, LNCS 948, pp. 456--468. Berlin, Springer-Verlag. MR1448185 (98g:12005)
26. Valibouze, A. (1999). Étude des relations algébriques entre les racines d'un polynôme d'une variable. Bull. Belgian Math. Soc. Simon Stevin, 6, 507--535. MR1732887 (2001b:12005)
27. Yokoyama, K. (1996). A modular method for computing the Galois groups of polynomials. In Cohen, A. and Roy, M., eds, Special Issue of MEGA '96, J. Pure Appl. Algebra, 117 \& 118, 617--636 MR1457858 (98h:12005)
28. Zariski, O., Samuel, P. (1967). Commutative Algebra, volume I. van Nostrand. MR0090581 (19,833e)

MR1737231 (2001e:12002)
Rennert, Nicolas(F-PARIS6-IP6); Valibouze, Annick(F-PARIS6-IP6)
Calcul de résolvantes avec les modules de Cauchy. (French. English, French summary) [Computing resolvents using Cauchy modules]
Experiment. Math. 8 (1999), no. 4, 351--366.
12F10

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The authors give a new and efficient algorithm to compute some characteristic polynomials of endomorphisms using Cauchy modules. This algorithm is used for the computation of absolute resolvents and multi-resolvents which are essential in the theory of fields and in the constructive Galois theory.

Reducing the time of computation of a resolvent improves the computation of the Galois group of a polynomial. This work presents a simple and quick algorithm (Theorem 4.7) which computes, without parasite factors, the characteristic polynomial of a multiplicative endomorphism associated with one given invariant. This characteristic polynomial is an exponent of the resolvent for the same invariant. The problem of the parasite factors is then solved, the parasite exponents; only the problem of the parasite exponents remains.

$§1$ and $§2$ present the introduction and give the notations. $§3$ recalls other results from the second author with or without J.-M. Arnaudiès.

In $§4$ the study of the characteristic polynomial in the algebra of universal decomposition reminds one of the methods of Arnaudiès (1992), J.-L. Lagrange (1770) and A. Cauchy (1882). The Cauchy modules, associated with the polynomial $f$, are $n$ polynomials if the degree of $f$ is $n$. One main result is (Theorem 4.7) an explicit algorithm giving the characteristic polynomial. And $(§5)$ if $f$ is separable, the Cauchy modules form a reduced Gröbner basis of $f$ from the ideal of the symmetric relations. The authors generalise this result to the case when $f$ is reducible.

In $§6$, devoted to the absolute resolvent (another main result), one uses the method of F. Lebohey (1997) for giving a computation algorithm of an exponent of the resolvent and next an algorithm of the resolvent with reductions. In $§7$ one finds the algorithm for an absolute multi-resolvent of $p$ polynomials.

At last $§8$ and $§9$ give implementation methods of different algorithms (with the system of the axiom formal calculus) and allow one to compare the efficiency.

This work is historically interesting, gives explicit algorithms and is relatively complete.

Reviewed by Jean-Daniel Thérond
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MR1732887 (2001b:12005)
Valibouze, Annick(F-PARIS6-IP6)
Étude des relations algébriques entre les racines d'un polynôme d'une variable. (French. English, French summary) [Study of the algebraic relations among the roots of a univariate polynomial]
Bull. Belg. Math. Soc. Simon Stevin 6 (1999), no. 4, 507--535.
12F10 (12Y05)

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The paper contains a detailed study of the relations among the roots of a polynomial by considering particular ideals containing the ideal consisting of all symmetric relations of the roots and contained in the (maximal) ideal of all relations between the roots. The main results of the paper are a correspondence between the above ideals and certain sets of permutations as well as an algorithm allowing the construction of a system of generators of the ideal of relations among the roots of the polynomial. The algorithm is illustrated by an explicit numerical example.

Reviewed by Chr. U. Jensen
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[References]

Note:

This list, extracted from the PDF form of the original paper, may contain data conversion errors, almost all limited to the mathematical expressions.

1. Abdeljaouad, I., 1997, Calcul d'invariants primitifs de groupes finis, Rapport LIP6 1997-020, `a paraître dans RAIRO-Informatique Théorique et Applications.
2. Anai H., Noro M., Yokoyama K., 1994 Computation of the splitting field and the Galois groups of polynomials, Progress in Mathematics, 143 (conference MEGA'94), Birkhaüser Verlag, pp. 29--50. MR1414444 (98a:12003)
3. Arnaudi`es J.M., Valibouze A., 1993. Résolvantes de Lagrange, Rapport LITP 93.61.
4. Arnaudi`es J.M., Valibouze A., 1996. Lagrange resolvents, special issue of MEGA'96 (A. Cohen and M-F- Roy Eds), Journ. of Pure and Appl. Algeb. 117&118 (1997), 23-40. MR1457831 (98f:12004)
5. Artin, E., 1959 Galois Theory, Notre Dame Mathematical Lectures No. 2, Notre Dame, IN : Notre Dame University Press. MR0265324 (42 #234)
6. Aubry, P., Valibouze, A., 1998, Using Galois ideals for computing relative resolvents, International Conference MEGA'98 (LIP6 Report 1998/004).
7. Berwick, E.H., 1915, The condition that a quintic equation should be soluble by radicals, Proc. London Math. Soc. (2) 14. 301-307.
8. Berwick, E.H., 1929 On Soluble sextic equations, Proc. London Math. Soc. (2) 29, 1-28.
9. Cauchy, A. Usage des fonctions interpolaires dans la détermination des fonctions symétriques des racines d'une équation algébrique donnée. \OE uvres Volume 5 p.473 extrait 108.
10. Colin, A., 1995 Formal computation of Galois groups with relative resolvents Conference AAECC'10, (Paris, July 1995), LNCS 948, 169-182 MR1448163 (98e:12001)
11. Colin A., 1997 Identification of the Galois group thanks to symbolic computation of relative resolvents and tables of partitions, ISSAC'97 Conference (Hawaii, July 1997).
12. Faug`ere, J.C., 1997 A new efficient algorithm for computing Gröbner Basis (F4), Task 3.3.2.1 Frisco report, preprint.
13. Foulkes, H.O., 1931, The resolvents of an equation of seventh degree, Quart. J. Math. Oxford Ser. (2), 9-19.
14. Galois, E., 1897 \OE uvres Math\'ematiques, publiées sous les auspices de la SMF, Gauthier-Villars.
15. GAP Groups, Algorithms and Programming, GAP 3.3 Martin Schönert and others, Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, 93
16. Girstmair, K., 1987 On invariant polynomials and their application in field theory Maths of Comp., vol. 48, no 178, 781-797. MR0878706 (88i:12001)
17. Kemper, G., 1996, Calculating invariant rings of finite groups over arbitrary fields, JSC 21 351-366. MR1400337 (98f:13005)
18. Kemper, G., 1994 The Invar package for calculating rings of invariants, IWR Preprint 93-34, Heidelberg.
19. Lagrange, J.L., 1770, Réflexions sur la résolution algébrique des équations, Mémoires de l'Académie de Berlin, 205-421, ( \OE uvres de Lagrange, tome IV, 205-421)
20. Lazard, D,, Valibouze, A., 1991, Computing subfields : Reverse of the primitive element problem, proc. de Mega'92 (Nice, april 1992). Progress in Mathematics 109, 163-176. MR1230864 (94h:68101)
21. McKay, J,, Soicher, L., 1985, Computing Galois Groups over the rationals, Journal of number theory 20, 273-281. MR0797178 (87a:12002)
22. Stauduhar, R.P., 1973, The determination of Galois groups, Math. Comp. 27, 981-996. MR0327712 (48 #6054)
23. Tchebotarev N., 1950 Grundz\"uge des Galois'shen Theorie, P. Noordhoff.
24. Valibouze A., 1989, Symbolic computation with symmetric polynomials, an extension to Macsyma. Conference Computers and Mathematics (1989, MIT, Cambridge, Mass.). Springer-Verlag, 308-320.
25. Valibouze A., 1995, Computation of the Galois group of the resolvent factors for the direct and inverse Galois problems, AAECC'10 conference (Paris, July 1995), LNCS 948, 456-468 (LITP Report 94-58). MR1448185 (98g:12005)
26. Yokoyama, K., Noro, M., Takeshima, T. , 1992 Solution of systems of algebraic equations and linear maps on Residue Class ring, Journal of Symbolic Computation 14, 399-417 . MR1187241 (94g:13020)

MR1457831 (98f:12004)
Arnaudiès, Jean-Marie(F-PARIS6-MI); Valibouze, Annick(F-PARIS6-C)
Lagrange resolvents. (English. English summary)
Algorithms for algebra (Eindhoven, 1996).
J. Pure Appl. Algebra 117/118 (1997), 23--40.
12F10

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This well-written paper deals with the determination of the Galois group of a given polynomial $f(X)\in K[X]$. Already Lagrange introduced resolvents to attack this problem. Several years ago Soicher, Foulkes, Stauduhar, McKay and others gave algorithms on how to compute Galois groups of polynomials of low degree $({\le}11)$ using resolvents.

This computation goes as follows: Given an irreducible polynomial $f(X)\in K[X]$ of degree $n$, first determine all transitive subgroups of ${\germ S}_n$ (up to conjugacy). Next choose some $F\in Z[X_1,\cdots,X_n]$. Denote the natural action of an element $\sigma\in{\germ S}_n$ on $F$ by $^\sigma F$. Let ${\scr O}(F)=\{^\sigma F, \sigma\in{\germ S}_n\}$ be the orbit of $F$. Then ${\rm Res}(F)=\prod_{G\in{\scr O}(F)}(X-G)$ is called the resolvent polynomial of $F$. Let $\alpha_1,\cdots,\alpha_n$ be the roots of $f(X)$ (in a fixed algebraic closure of $K)$. Next factor ${\rm Res}(F,f)\coloneq{\rm Res}(F)(\alpha_1,\cdots,\alpha_n)$ over $K$. If ${\rm Res}(F,f)$ is separable, then the partition of the degree of ${\rm Res}(F,f)$ given by the degrees of the irreducible factors of ${\rm Res}(F,f)$ coincides with the partition of $\deg({\rm Res}(F))=\#({\scr O}(F))$ given by the lengths of the orbits obtained by the action of ${\rm Gal}(f)$ on ${\scr O}(F)$. It may happen that it is not possible to deduce the Galois group from the factorization of a single separable resolvent. For example, the factorization of the discriminant of $f(X)$ tells us only whether ${\rm Gal}(F)$ is contained in ${\germ A}_n$ or not. Therefore, we have to factor another resolvent. In the literature we find several tables of resolvents whose factorizations suffice to compute the Galois group of a polynomial of small degree.

Let $K$ be a field of characteristic 0. The main theorem of the paper under review states that it is always possible to compute the Galois group of $f(X)$ from the factorization of a finite number of separable resolvents. The authors call it the chasing resolvents method.

It runs as follows: A partition of an integer $m\inN$ is an $m$-tuple $(\alpha_1,\cdots,\alpha_m)\inN_0^m$ with $\sum_{j=1}^mj\alpha_j=m$. Let $U,V$ be subgroups of $G$ and set $e\coloneq[G\:U]$. Let ${\scr P}(V,U)=(\alpha_1,\cdots,\alpha_e)$ be the partition given by the orbit lengths of the action of $V$ on $G/U$ (i.e. $\alpha_j$ is the number of orbits of length $j)$. Now ${\scr P}(V,U)$ depends only on the conjugacy classes of $U$ and $V$ in $G$. Let ${\scr C}_1,\cdots,{\scr C}_s$ be the conjugacy classes of subgroups of $G$ (with a given ordering). Set $\overline\omega({\scr C}_i,{\scr C}_j)={\scr P}(V,U)$ where $V\in{\scr C}_i, U\in{\scr C}_j$. Then ${\scr P}_G=[\overline\omega({\scr C}_i,{\scr C}_j)]_{1\le i,j\le s}$ is called the partition matrix of $G$. Theorem 14 states that the rows of this matrix are distinct.

Next we have to introduce the resolvents. Set ${\scr F}=K(X_1,\cdots,X_n), {\scr K}=K(\sigma_1,\cdots,\sigma_n)$, where $\sigma_i$ is the $i{\rm th}$ elementary symmetric polynomial in $X_1,\cdots,X_n$. Let $H<{\germ S}_n$. Let $\Psi$ be a polynomial in the integral closure ${\scr F}^H\cap K[\sigma_1,\cdots,\sigma_n]$ of ${\scr S}=K[\sigma_1,\cdots,\sigma_n]$ in the fix field ${\scr F}^H$ of $H$ which is a primitive element of the extension ${\scr F}/{\scr K}$. Then the minimal polynomial ${\scr L}_\Psi$ over ${\scr K}$ is called the (generic) Lagrange resolvent of $H$ associated to $\Psi$. Assume that ${\scr L}_{\Psi,f}={\scr L}_\Psi(\alpha_1,\cdots,\alpha_n)$ is separable, where $f(X)=\prod_{i=1}^n(X-\alpha_i)$. Then the partition given by the degrees of the irreducible factors of ${\scr L}_{\Psi,f}$ over $K$ coincides with ${\scr P}_{{\germ S}_n}(G,H)$, where $G={\rm Gal}(f)$. Since the rows of ${\scr P}_{{\germ S}_n}$ are distinct we are done. Hence the conjugacy classes in the columns play the role of test classes. The candidates for the Galois group can be found in the rows of the partition matrix.

The authors illustrate their approach with an example. They compute ${\scr P}_{{\germ S}_4}$ and give Lagrange resolvents. From this example we see that we do not have to consider all resolvents. If $n=4$, then two resolvents instead of 11 suffice. Section 4 contains an extension of the chasing resolvents method to relative resolvents. This gives a way of throwing out multiple factors in the absolute resolvent.

{For the entire collection see MR1457829 (97m:00022).}

Reviewed by Martin Epkenhans
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MR1448185 (98g:12005)
Valibouze, Annick(F-PARIS6-C)
Computation of the Galois groups of the resolvent factors for the direct and inverse Galois problems. (English. English summary)
Applied algebra, algebraic algorithms and error-correcting codes (Paris, 1995), 456--468,
Lecture Notes in Comput. Sci., 948,
Springer, Berlin, 1995.
12F10 (12F12)

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Let $k$ be a field of characteristic 0 and $f(x)$ a squarefree polynomial of degree $n$. Let $a_1,\cdots ,a_n$ be the roots of $f(x)$ in some extension field of $k$ and let $H$ be a subgroup of $S_n$ which acts in the obvious way on $k[x_1,\cdots ,x_n]$. Let $h[x_1,\cdots ,x_n]$ be a polynomial which is also a primitive element for $k(x_1,\cdots ,x_n)^H$ over $k(x_1,\cdots ,x_n)^{S_n}$. Let $\operatorname{id}=f_1,f_2,f_e$ be a set of representatives of the left cosets of $H$ in $S_n$ and let $h_i(x_1,\cdots ,x_n)=f_i(h(x_1,\cdots ,x_n))$. The resolvent of $f(x)$ by $h(x_1,\cdots ,x_n)$ is the univariate polynomial $(y-h_1(a_1,\cdots ,a_n))\cdots (y-h_e(a_1, \cdots ,a_n))$. The Galois group of the resolvents can be used to calculate the Galois group of the original polynomial $f(x)$. This paper presents results for calculating the Galois groups of resolvents for polynomials of low degree, particularly degrees 8 and 10.

{For the entire collection see MR1448151 (97k:68003).}

Reviewed by James K. Deveney
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MR1230864 (94h:68101)
Lazard, D.(F-PARIS6-C); Valibouze, A.(F-PARIS6-C)
Computing subfields: reverse of the primitive element problem. (English. English summary)
Computational algebraic geometry (Nice, 1992), 163--176,
Progr. Math., 109,
Birkhäuser Boston, Boston, MA, 1993.
68Q40 (12Y05)

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A new algorithm is presented to compute the subextensions of some effectively given algebraic field extension $K\to L$, for arbitrary base field $K$. Similarly, the algorithm can compute low degree subfields of low degree extensions of $L$, but in general the Galois group or Galois closure will not be computed. Some examples where $K$ is the field of rational numbers are given. No runtime analysis is given, and it is not clear how the efficiency of the algorithm compares to older algorithms for finding subfields. Such algorithms are useful for the simplification of algebraic numbers.

{For the entire collection see MR1230853 (94b:13001).}

Reviewed by A. K. Lenstra
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MR1226584 (94g:05099)
Valibouze, Annick(F-PARIS6-C)
Sur l'arité des fonctions. (English. English summary) [On the arity of functions]
European J. Combin. 14 (1993), no. 4, 359--372.
05E05 (12E99 20B30)

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The author gives an approach to the computation of resolvents of polynomials. Motivated by the process of extending $S\sb m$-symmetry to $S\sb n$-symmetry of polynomials in $n$ variables $(m\leq n)$ an analogue is considered for certain permutation subgroups of $S\sb n$. Applications have been implemented into the computer algebra system MACSYMA.

Reviewed by Thomas Scharf
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MR1034742 (91a:13013)
Giusti, Marc(F-POLY); Lazard, Daniel(F-PARIS6-C); Valibouze, Annick(F-CNRS-L)
Algebraic transformations of polynomial equations, symmetric polynomials and elimination.
Symbolic and algebraic computation (Rome, 1988), 309--314,
Lecture Notes in Comput. Sci., 358,
Springer, Berlin, 1989.
13P05 (14Q99)

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Consider the following problems: (i) Calculate the polynomial whose roots are the squares of a given polynomial. (ii) Given two polynomials $P$ and $Q$, calculate the polynomial whose roots are all the differences of a root of $P$ and a root of $Q$.

These are examples of the problem of "transforming equations by a morphism". In this paper it is shown (after a technical definition of the concept) that "transforming polynomial equations by a morphism" is algorithmically equivalent to elementary elimination theory by resultants and also to making change of bases for symmetric polynomials. Implementations in computer algebra systems and efficiency are discussed.

{For the entire collection see MR1034718 (90i:00005).}

Reviewed by Ralf Fröberg
Previous Item

MR1033311 (90m:05006)
Valibouze, Annick(F-PARIS6-C)
Fonctions symétriques et changements de bases. (French. English summary) [Symmetric functions and changes of basis]
EUROCAL '87 (Leipzig, 1987), 323--332,
Lecture Notes in Comput. Sci., 378,
Springer, Berlin, 1989.
05-04 (12-04 68R05)

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This paper describes two change-of-basis algorithms: the first one expresses a symmetric polynomial in terms of elementary symmetric polynomials, the second one in terms of power sums. They proceed by using a decomposition formula for the product of two monomial polynomials, which is triangular with respect to two appropriate total orders on the partitions. The computations are made formally from a representative for the monomial polynomials; this avoids having to develop the monomial polynomials and saves a lot of time.

{For the entire collection see MR1033288 (90i:68007).}

Reviewed by Laurent Habsieger
Previous Item