Geodesic
Generalized typical dimension of a~graded module
Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 2, pp. 147-161.

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In this paper, we prove an upper bound for the leading coefficient of the characteristic polynomial of a graded ideal in a ring of generalized polynomials. Examples of such rings are the rings of commutative polynomials (for which the classical Bézout theorem holds), as well as some rings of differential operators. For a system of generalized homogeneous equations in small codimensions we obtain exact estimates that are polynomial in $d$. In the general case, the estimate is double exponential in $\tau$: $O\bigl(d^{2^{\tau-1}}\bigr)$, where $d$ is the maximal degree of generators of a graded ideal and $\tau$ is its codimension. For systems of linear differential equations, bounds of the same asymptotics, but by other methods, were obtained by D. Grigoriev. 
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M. V. Kondratieva. Generalized typical dimension of a~graded module. Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 2, pp. 147-161. http://geodesic.mathdoc.fr/item/FPM_2020_23_2_a7/

[1] Kondrateva M. V., “Opisanie mnozhestva minimalnykh differentsialnykh razmernostnykh mnogochlenov”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1988, no. 1, 35–39 | Zbl

[2] Kondrateva M. V., “Otsenka tipovoi differentsialnoi razmernosti sistemy lineinykh differentsialnykh uravnenii”, Fundament. i prikl. matem., 22:5 (2019), 259–269 | MR

[3] Dubé T., “The structure of polynomial ideals and Gröbner bases”, SIAM J. Comput., 19:4 (1990), 750–773 | DOI | MR | Zbl

[4] Grigoriev D., “Weak Bezout inequality for D-modules”, J. Complexity, 21 (2005), 532–542 | DOI | MR | Zbl

[5] Chistov A., Grigoriev D., “Complexity of a standard basis of a D-module”, St. Petersburg Math. J., 20 (2009), 709–736 | DOI | MR | Zbl

[6] Kolchin E. R., Differential Algebra and Algebraic Groups, Academic Press, 1973 | MR | Zbl

[7] Kondratieva M. V., Levin A. B., Mikhalev A. V., Pankratiev E. V., Differential and Difference Dimension Polynomials, Kluwer Academic, 1999 | MR | Zbl

[8] Mayr E. W., Meyer A. R., “The complexity of a word problem for commutative semigroups and polynomial ideals”, Adv. Math., 46 (1982), 305–329 | DOI | MR | Zbl

[9] Ritt J., Differential Algebra, Amer. Math. Soc., New York, 1950 | MR | Zbl