Abstract
In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Letk be an infinite perfect field and let f1,...,f n−r∈k[X1,...,Xn] be a regular sequence with d:=maxj deg fj. Denote byA the polynomial ringk [X1,..., Xr] and byB the factor ring k[X1,...,Xn]/(f1,...,fn r); assume that the canonical morphism A→B is injective and integral and that the Jacobian determinantΔ with respect to the variables Xr+1,...,Xn is not a zero divisor inB. Let finally σ∈B*:=HomA(B, A) be the generator of B* associated to the regular sequence.
We show that for each polynomialf the inequality deg σ(¯f) ≦dn r(δ+1) holds (¯fdenotes the class off inB andδ is an upper bound for (n−r)d and degf). For the usual trace associated to the (free) extensionA ↪B we obtain a somewhat more precise bound: deg Tr(¯f) ≦ dn r degf. From these bounds and Bertini's theorem we deduce an elementary proof of the following effective Nullstellensatz: let f1,..., fs be polynomials in k[X1,...,Xn] with degrees bounded by a constant d≧2; then 1 ∈(f1,..., fs) if and only if there exist polynomials p1,..., ps∈k[X1,..., Xn] with degrees bounded by 4n(d+ 1)n such that 1=Σipifi. in the particular cases when the characteristic of the base fieldk is zero ord=2 the sharper bound 4ndn is obtained.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Becker, E., Wörman, T.: On the trace formula for quadratic forms and some applications. To appear in Proc. RASQUAD, Berkeley
Berenstein, C., Struppa, D.: Recent improvements in the Complexity of the Effective Nullstellensatz. Linear Algebra Its Appl.157, 203–215 (1991)
Berenstein, C., Yger, A.: Bounds for the degrees in the division problem. Mich. Math. J.37, 25–43 (1990)
Brownawell, D.: Bounds for the degrees in the Nullstellensatz. Ann. Math. Second Series126(3), 577–591 (1987)
Caniglia, L., Galligo, A., Heintz, J.: Some new effectivity bounds in computational geometry. Proc. 6th Int. Conf. Applied Algebra, Algebraic Algorithms and Error Correcting Codes AAECC-6, Roma 1988, Lecture Notes in Computer Sciences. Berlin, Hiedelberg, New York: Springer vol.357, 131–151 (1989)
Cardinal, J.-P.: Dualité et algorithmes itératifs pour la résolution de systèmes polynomiaux. Thesis Université de Rennes (1993)
Dickenstein, A., Giusti, M., Fitchas, N., Sessa, C.: The membership problem for unmixed polynomial ideals is solvable in single exponential time. Discrete Appl. Math.33, 73–94 (1991)
Dickenstein, A., Sessa, C.: An effective residual criterion for the membership problem in ℂ[z1,..., zn]. J. Pure Appl. Algebra74, 149–158 (1991)
Dubé, T.: A Combinatorial Proof of the Effective Nullstellensatz. J. Symb. Comp.15, 277–296 (1993)
Fitchas, N., Galligo, A.: Nullstellensatz effectif et conjecture de Serre (théorème de Quillen-Suslin) pour le Calcul Formel. Math. Nachr.149, 231–253 (1990)
Fitchas, N., Giusti, M., Smietanski, F.: Sur la complexité du théorème des zéros. Preprint Ecole Polytechnique Palaiseau (1992)
Giusti, M., Heintz, J.: La détermination des points isolés et de la dimension d'une variété algébrique peut se faire en temps polynomial. To appear in Proc. Int. Meeting on Commutative Algebra, Cortona, 1991
Giusti, M., Heintz, J., Sabia, J.: On the efficiency of effective Nullstellensätze. Comput. Complexity3, 56–95 (1993)
Heintz, J.: Definability and fast quantifier elimination in algebraically closed fields. Theoret. Comput. Sci.24, 239–277 (1983)
Iversen, B.: Generic Local Structures in Commutative Algebra. Lect. Notes in Math. vol.310. Berlin, Heidelberg, New York: Springer 1973
Jouanolou, J.-P.: Théorèmes de Bertini et applications. Progress in Math. vol.42. Basel: Birkhäuser (1983)
Kreuzer, M., Kunz, E.: Traces in strict Frobenius algebras and strict complete intersections. J. Reine Angew. Math.381, 181–204 (1987)
Kollár, J.: Sharp effective Nullstellensatz. J. AMS1, 963–975 (1988)
Kunz, E.: Kälher Differentials. Adv. Lect. in Math. Vieweg Verlag (1986)
Logar, A.: A computational proof of the Noether normalization lemma. Proc. 6th Int. Conf. Applied Algebra, Algebraic Algorithms and Error Correcting Codes AAECC-6, Roma 1988, Lecture Notes in Computer Sciences vol.357, pp. 259–273. Berlin, Heidelberg, New York: Springer 1989
Matsumura, H.: Commutative Algebra. Benjamin (1970)
Matsumura, H.: Commutative ring theory. Cambridge Studies in Adv. Math. vol.8. Cambridge University Press (1989)
Mumford, D.: The Red Book of Varieties and Schemes. Lect. Notes in Math. vol.1358. Berlin, Heidelberg, New York: Springer 1988
Pedersen, P., Roy, M.-F., Szpirglas, A.: Counting real zeros in the multivariate case. To appear in Proc. MEGA 92
Scheja, G., Storch, U.: Über Spurfunktionen bei vollständigen Durchschnitten. J. Reine Angew. Math.278, 174–190 (1975)
Teissier, B.: Résultats récents d'algèbre commutative effective. Séminaire Bourbaki 1989–1990, Astérisque vol.189–190, 107–131 (1991)
Vasconcelos, W.: Jacobian Matrices and Constructions in Algebra. Proc. 9th Int. Conf. Applied Algebra, Algebraic Algorithms and Error Correcting Codes AAECC-9, New Orleans, 1991, LN Comput. Sci. vol.539, pp. 48–64. Berlin, Heidelberg, New York: Springer 1992
Author information
Authors and Affiliations
Additional information
Partially supported by UBACYT and CONICET (Argentina)
Rights and permissions
About this article
Cite this article
Sabia, J., Solernó, P. Bounds for traces in complete intersections and degrees in the Nullstellensatz. AAECC 6, 353–376 (1995). https://doi.org/10.1007/BF01198015
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01198015