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.

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).}

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

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

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).}

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