Resultant, discriminant
The paper under review is a revised version of a text on the elementary theory of resultants and discriminants intended as an appendix to Chapter IV (``Polynomials and Rational Fractions'') of Bourbaki's treatise on ``Algebra''. The Bourbaki ``congress'' held in july 1969 decided to abandon the project, mainly because some of the results used in the text, like Gauss' lemma for example, depended on the notion of factorial ring which appears only in Chapter VII of Bourbaki's book on ``Commutative Algebra''. Compared to the original version, the paper under review contains some modifications and the addition of two appendices: one on Gauss' lemma and the other one on Hilbert's Nullstellensatz. After this information, extracted from the Introduction of the paper, let us briefly describe its content. In Section 1 one proves the basic result of elimination theory. Let \(A\) be an (always commutative) ring and \(I\) a homogeneous ideal of the polynomial ring \(A[X_1,\dots ,X_n]\). The \textit{eliminating ideal} of \(I\) is the ideal \(\mathfrak{e}(I)\) of \(A\) defined by: \[ \mathfrak{e}(I):= \{a\in A\, | \, aX_i^m \in I\;\text{for}\;1\leq i \leq n \;\text{and}\;m >> 0\}\, . \] If \(k\) is a field and \(\rho: A \rightarrow k\) is a morphism of rings then the \textit{Elimination Theorem} asserts that \(\rho(\mathfrak{e}(I)) = (0)\) if and only if there exists a field extension \(K\) of \(k\) such that the homogeneous elements of \(I\) have a non-trivial common zero in \(K^n\). Actually, if \(\rho(\mathfrak{e}(I)) = (0)\) then one can find such a \(K\) of finite degree over \(k\). In Section 2 one introduces some notation. For fixed integers \(n\geq 1\), \(r\geq 1\), \(d_1\geq 1,\dots ,d_r\geq 1\), one considers the \textit{universal} homogeneous polynomials in \(n\) indeterminates, of degrees \(d_1,\dots ,d_r\), respectively: \[ P_j = { \sum_{| \alpha | = d_j}}T_{\alpha,j}X^\alpha ,\;j = 1,\dots ,r\, , \] where, for \(\alpha = (a_1,\dots ,a_n)\in {\mathbb N}^n\), one denotes by \(X^\alpha\) the monomial \(X_1^{a_1}\cdots X_n^{a_n}\) and \(| \alpha | = a_1 + \cdots + a_n\). One also considers the polynomial \(\mathbb Z\)-algebra \(U = U_{n;d_1,\dots ,d_r} = {\mathbb Z}[(T_{\alpha,j})]\) generated by their coefficients. If \(A\) is a ring and if \(f_1,\dots ,f_r\) are homogeneous polynomials in \(A[X_1,\dots ,x_n]\), of degrees \(d_1,\dots ,d_r\), respectively, then, for \(u\in U\), one denotes by \({\widetilde u}(f_1,\dots ,f_r)\) the element of \(A\) obtained by replacing, in \(u\), the indeterminate \(T_{\alpha,j}\) by the coefficient of \(X^\alpha\) in \(f_j\). In Section 3 one shows that if \(I\) is the ideal of \(U[X_1,\dots ,X_n]\) generated by \(P_1,\dots ,P_r\) then \(\mathfrak{e}(I)\) is a prime ideal of \(U\) which is non-zero if and only if \(r\geq n\). Moreover, for \(r = n\), \(\mathfrak{e}(I)\) is a principal ideal of \(U\). In Section 4 one assumes \(r = n\) and one denotes by \(R\) the unique generator \(u\) of \(\mathfrak{e}((P_1,\dots ,P_n))\) with the property that \({\widetilde u}(X_1^{d_1},\dots , X_n^{d_n}) = 1\). For any ring \(A\) and for any homogeneous polynomials \(f_1,\dots , f_n\) in \(A[X_1,\dots ,X_n]\) of degrees \(d_1,\dots , d_n\), respectively, one defines the \textit{resultant} \(\text{res}(f_1,\dots ,f_n)\) as the element \({\widetilde R}(f_1,\dots ,f_n)\) of \(A\). One proves, then, the main properties of the resultant. In particular, one shows that \(R\) is a greatest common divisor of the \textit{Sylvester determinants}. In Section 5 one defines the discriminant of a homogeneous polynomial and one proves its basic properties. Let \(n\geq 1\) and \(d \geq 2\) be two integers, let \(P_{n,d} = \sum_{| \alpha | = d}T_\alpha X^\alpha\) be the universal homogeneous polynomial of degree \(d\) in \(n\) indeterminates and let \(U_{n,d}\) denote the polynomial ring \({\mathbb Z}[(T_\alpha)_{| \alpha | = d}]\). One shows, firstly, that if \(J(P_{n,d})\) is the ideal of \(U_{n,d}[X_1,\dots , X_n]\) generated by \(P_{n,d}\) and by its first order partial derivatives then \(\mathfrak{e}(J(P_{n,d}))\) is a prime ideal of \(U_{n,d}\). Then, one proves that the \textit{content} (= g.c.d. of the coefficients) of \(\text{res}(\partial P_{n,d}/\partial X_1,\dots , \partial P_{n,d}/\partial X_n)\) is \(d^{a(n,d)}\), where \(a(n,d) = \frac{(d-1)^n-(-1)^n}{d}\). One defines, now, the \textit{universal divided discriminant} by: \[ \text{disc}(P_{n,d}):= \frac{1}{d^{a(n,d)}} \text{res}(\partial P_{n,d}/\partial X_1,\dots ,\partial P_{n,d}/\partial X_n) \in U_{n,d}\, . \] If \(A\) is a ring and if \(f\) is a homogeneous polynomial of degree \(d\) in \(A[X_1,\dots ,X_n]\) then one puts \(\text{disc}(f):= \text{disc}(P_{n,d})\sptilde(f)\). One of the main results of this section asserts that, if \(A\) is a field \(k\), there exists a field extension \(L\) of \(k\) such that \(f\) admits a non-trivial \textit{critical} zero in \(L^n\) if and only if \(\text{disc}(f) = 0\) and, if this is the case, one can find such an \(L\) of finite degree over \(k\). In the final Section 6 one considers the problem of the irreducibility of the discriminant. One proves that if \(K\) is a field whose characteristic divides neither \(d\) nor \(d - 1\), then \(\text{disc}(1_K\otimes P_{n,d}) \in K\otimes_{\mathbb Z}U_{n,d}\) is irreducible and generates \(\mathfrak{e}(J(1_K\otimes P_{n,d}))\). The proof uses a lemma asserting that if \[ P(X_1,\dots ,X_n):= { \sum_{i=1}^n}X_i^d - (Z_1X_1 + \cdots + Z_nX_n)^d \in A[X_1,\dots ,X_n]\, , \] where \(A = K[Z_1,\dots ,Z_n]\), then \(\text{disc}(P) \in K[Z_1,\dots ,Z_n]\) is irreducible (under the above hypothesis on \(\text{char}\, K\)). The proof of this lemma is rather involved, although elementary. Actually, the irreducibility of \(\text{disc}(1_K\otimes P_{n,d})\) fails only if \(\text{char}\, K = 2\) and \(n\) is even, but for the proof of this fact the author refers the reader to the paper of \textit{T. Saito} [Math. Res. Lett. 19, No. 4, 855--871 (2012; Zbl 1285.14046)].
scientific article; zbMATH DE number 6149827