Equations and Complexity for the Dubois–Efroymson Dimension Theorem

Ghiloni, Riccardo

Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 224-236

Résumé

Let $R$ be a real closed field, let $X\,\subset \,{{R}^{n}}$ be an irreducible real algebraic set and let $Z$ be an algebraic subset of $X$ of codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset of $X$ of codimension 1 containing $Z$ . We improve this dimension theorem as follows. Indicate by $\mu$ the minimum integer such that the ideal of polynomials in $R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ vanishing on $Z$ can be generated by polynomials of degree $\le \,\mu$ . We prove the following two results: (1) There exists a polynomial $P\,\in \,R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $\le \,\mu +1$ such that $X\cap {{P}^{-1}}\left( 0 \right)$ is an irreducible algebraic subset of $X$ of codimension 1 containing $Z$ . (2) Let $F$ be a polynomial in $R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $d$ vanishing on $Z$ . Suppose there exists a nonsingular point $x$ of $X$ such that $F\left( x \right)\,=\,0$ and the differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then there exists a polynomial $G\,\in \,R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $\le \,\max \{d,\,\mu \,+\,1\}$ such that, for each $t\,\in \,\left( -1,\,1 \right)\,\backslash \,\{0\}$ , the set $\{x\in X|F\left( x \right)+tG\left( x \right)=0\}$ is an irreducible algebraic subset of $X$ of codimension 1 containing $Z$ . Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

Détail

DOI : 10.4153/CMB-2009-025-9

Mots-clés : 14P05, 14P20, Irreducible algebraic subvarieties, complexity of algebraic varieties, Bertini's theorems

Bibliographie

[1] [1] Dubois, D. and Efroymson, G., A dimension theorem for real primes. Canad. J. Math. 26(1974), 108–114. Google Scholar

[2] [2] Jouanolou, J.-P., Théorèmes de Bertini et applications. Progress inMathematics 42, Birkhaüser Boston, Inc., Boston, MA, 1983. Google Scholar

[3] [3] Kucharz, W., A note on the Dubois–Efroymson dimension theorem. Canad. Math. Bull. 32(1989), no. 1, 24–29. Google Scholar

[4] [4] Mumford, D., Algebraic geometry I. Complex projective varieties. Grundlehren der Mathematischen Wissenschaften 221, Springer–Verlag, Berlin-New York, 1976. Google Scholar

[5] [5] Serre, J.-P., Faisceaux algébriques cohérents. Ann. of Math. (2) 61(1955), 197–278. Google Scholar

[6] [6] Shafarevich, I. R., Basic algebraic geometry 1. Varieties in projective space. Second edition, Springer-Verlag, Berlin, 1994. Google Scholar

[7] [7] Whitney, H., Elementary structure of real algebraic varieties. Ann. of Math. (2) 66(1957), 545–556. Google Scholar