{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T09:55:51Z","timestamp":1776074151381,"version":"3.50.1"},"reference-count":63,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s00200-026-00726-9","type":"journal-article","created":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T15:04:33Z","timestamp":1775055873000},"page":"207-304","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Kronecker algorithm for locally closed sets over a perfect field"],"prefix":"10.1007","volume":"37","author":[{"given":"Nardo","family":"Gim\u00e9nez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joos","family":"Heintz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guillermo","family":"Matera","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luis Miguel","family":"Pardo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mariana","family":"P\u00e9rez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Melina","family":"Privitelli","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,4,1]]},"reference":[{"key":"726_CR1","doi-asserted-by":"crossref","unstructured":"Abhyankar, S.S.: Algebraic Geometry for Scientists and Engineers, Math. Surv. Monogr., vol. 35. American Mathematical Society, Providence (1990)","DOI":"10.1090\/surv\/035"},{"key":"726_CR2","doi-asserted-by":"crossref","unstructured":"Alonso, M.E., Becker, E., Roy, M.-F., W\u00f6rmann, T.: Zeroes, multiplicities and idempotents for zerodimensional systems. In: Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA\u201994. Progr. Math., vol. 143, pp. 1\u201315. Birkh\u00e4user, Boston (1996)","DOI":"10.1007\/978-3-0348-9104-2_1"},{"key":"726_CR3","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1016\/0304-3975(83)90110-X","volume":"22","author":"W Baur","year":"1983","unstructured":"Baur, W., Strassen, V.: The complexity of partial derivatives. Theor. Comput. Sci. 22, 317\u2013330 (1983)","journal-title":"Theor. Comput. Sci."},{"issue":"2","key":"726_CR4","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1090\/S0894-0347-08-00630-9","volume":"22","author":"C Beltr\u00e1n","year":"2009","unstructured":"Beltr\u00e1n, C., Pardo, L.M.: Smale\u2019s 17th problem: average polynomial time to compute affine and projective solutions. J. Am. Math. Soc. 22(2), 363\u2013385 (2009)","journal-title":"J. Am. Math. Soc."},{"key":"726_CR5","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/s10208-010-9078-9","volume":"11","author":"C Beltr\u00e1n","year":"2011","unstructured":"Beltr\u00e1n, C., Pardo, L.M.: Fast linear homotopy to find approximate zeros of polynomial systems. Found. Comput. Math. 11, 95\u2013129 (2011)","journal-title":"Found. Comput. Math."},{"key":"726_CR6","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1016\/0020-0190(84)90018-8","volume":"18","author":"SJ Berkowitz","year":"1984","unstructured":"Berkowitz, S.J.: On computing the determinant in small parallel time using a small number of processors. Inf. Process. Lett. 18, 147\u2013150 (1984)","journal-title":"Inf. Process. Lett."},{"issue":"2\u20133","key":"726_CR7","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1016\/j.tcs.2004.01.015","volume":"315","author":"A Bompadre","year":"2004","unstructured":"Bompadre, A., Matera, G., Wachenchauzer, R., Waissbein, A.: Polynomial equation solving by lifting procedures for ramified fibers. Theor. Comput. Sci. 315(2\u20133), 335\u2013369 (2004)","journal-title":"Theor. Comput. Sci."},{"key":"726_CR8","doi-asserted-by":"crossref","unstructured":"B\u00fcrgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Grundlehren Math. Wiss., vol. 315. Springer, Berlin (1997)","DOI":"10.1007\/978-3-662-03338-8"},{"issue":"2","key":"726_CR9","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/j.ffa.2005.03.003","volume":"12","author":"A Cafure","year":"2006","unstructured":"Cafure, A., Matera, G.: Improved explicit estimates on the number of solutions of equations over a finite field. Finite Fields Appl. 12(2), 155\u2013185 (2006)","journal-title":"Finite Fields Appl."},{"issue":"256","key":"726_CR10","doi-asserted-by":"publisher","first-page":"2049","DOI":"10.1090\/S0025-5718-06-01878-3","volume":"75","author":"A Cafure","year":"2006","unstructured":"Cafure, A., Matera, G.: Fast computation of a rational point of a variety over a finite field. Math. Comput. 75(256), 2049\u20132085 (2006)","journal-title":"Math. Comput."},{"key":"726_CR11","unstructured":"Dahan, X., Moreno Maza, M., Schost, E., Xie, Y.: On the complexity of the D5 principle. In: Proceedings of Transgressive Computing 2006. A Conference in Honor of Jean Della Dora, Granada, Espa\u00f1a, April 24\u201326, 2006, pp. 149\u2013167. Universidad de Granada, Granada (2006)"},{"issue":"4","key":"726_CR12","first-page":"571","volume":"46","author":"C D\u2019Andrea","year":"2013","unstructured":"D\u2019Andrea, C., Krick, T., Sombra, M.: Heights of varieties in multiprojective spaces and arithmetic Nullstellens\u00e4tze. Ann. Sci. \u00c9c. Norm. Sup\u00e9r. (4) 46(4), 571\u2013649 (2013)","journal-title":"Ann. Sci. \u00c9c. Norm. Sup\u00e9r. (4)"},{"key":"726_CR13","series-title":"Encyclopaedia of Mathematical Sciences","first-page":"167","volume-title":"Algebraic Geometry I","author":"V Danilov","year":"1994","unstructured":"Danilov, V.: Algebraic varieties and schemes. In: Shafarevich, I.R. (ed.) Algebraic Geometry I. Encyclopaedia of Mathematical Sciences, vol. 23, pp. 167\u2013307. Springer, Berlin (1994)"},{"key":"726_CR14","doi-asserted-by":"crossref","unstructured":"Della Dora, J., Discrescenzo, C., Duval, D.: About a new method for computing in algebraic number fields. In: Caviness, B.F. (ed.) Proceedings of the 10th European Conference on Computer Algebra, EUROCAL \u201985. Vol.\u00a02: Research Contributions (Linz, Austria, April 1-3, 1985). Lecture Notes in Comput. Sci., vol. 204, pp. 289\u2013290. Springer, Berlin (1985)","DOI":"10.1007\/3-540-15984-3_279"},{"issue":"4","key":"726_CR15","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/0020-0190(78)90067-4","volume":"7","author":"RA De Millo","year":"1978","unstructured":"De Millo, R.A., Lipton, R.J.: A probabilistic remark on algebraic program testing. Inf. Process. Lett. 7(4), 193\u2013195 (1978)","journal-title":"Inf. Process. Lett."},{"issue":"2","key":"726_CR16","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.exmath.2007.07.001","volume":"26","author":"C Durvye","year":"2008","unstructured":"Durvye, C., Lecerf, G.: A concise proof of the Kronecker polynomial system solver from scratch. Expo. Math. 26(2), 101\u2013139 (2008)","journal-title":"Expo. Math."},{"key":"726_CR17","volume-title":"Commutative Algebra with a View Toward Algebraic Geometry. Grad. Texts in Math.,","author":"D Eisenbud","year":"1995","unstructured":"Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Grad. Texts in Math., vol. 150. Springer, New York (1995)"},{"key":"726_CR18","unstructured":"Fitchas, N., Giusti, M., Smietanski, F.: Sur la complexit\u00e9 du th\u00e9or\u00e8me des z\u00e9ros. In: Guddat et al, J. (ed.) Approximation and Optimization in the Caribbean II, Proceedings 2nd International Conference on Non-Linear Optimization and Approximation. Approximation and Optimization, vol. 8, pp. 247\u2013329. Peter Lange Verlag, Frankfurt am Main (1995)"},{"key":"726_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02421-8","volume-title":"Intersection Theory","author":"W Fulton","year":"1984","unstructured":"Fulton, W.: Intersection Theory. Springer, Berlin (1984)"},{"key":"726_CR20","volume-title":"Modern Computer Algebra","author":"J Gathen","year":"1999","unstructured":"Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)"},{"key":"726_CR21","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.jco.2018.09.005","volume":"51","author":"N Gim\u00e9nez","year":"2019","unstructured":"Gim\u00e9nez, N., Matera, G.: On the bit complexity of polynomial system solving. J. Complex. 51, 20\u201367 (2019)","journal-title":"J. Complex."},{"issue":"2","key":"726_CR22","doi-asserted-by":"publisher","first-page":"97","DOI":"10.4064\/aa220817-21-7","volume":"211","author":"N Gim\u00e9nez","year":"2023","unstructured":"Gim\u00e9nez, N., Matera, G., P\u00e9rez, M., Privitelli, M.: The distribution of defective multivariate polynomial systems over a finite field. Acta Arith. 211(2), 97\u2013120 (2023)","journal-title":"Acta Arith."},{"key":"726_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2022.101712","volume":"75","author":"N Gim\u00e9nez","year":"2023","unstructured":"Gim\u00e9nez, N., Matera, G., P\u00e9rez, M., Privitelli, M.: On the computation of rational solutions of underdetermined systems over a finite field. J. Complex. 75, 101712 (2023)","journal-title":"J. Complex."},{"key":"726_CR24","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/S0022-4049(97)00015-7","volume":"117118","author":"M Giusti","year":"1997","unstructured":"Giusti, M., H\u00e4gele, K., Heintz, J., Morais, J.E., Monta\u00f1a, J.L., Pardo, L.M.: Lower bounds for Diophantine approximation. J. Pure Appl. Algebra 117118, 277\u2013317 (1997)","journal-title":"J. Pure Appl. Algebra"},{"key":"726_CR25","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/S0022-4049(96)00099-0","volume":"124","author":"M Giusti","year":"1998","unstructured":"Giusti, M., Heintz, J., Morais, J.E., Morgenstern, J., Pardo, L.M.: Straight-line programs in geometric elimination theory. J. Pure Appl. Algebra 124, 101\u2013146 (1998)","journal-title":"J. Pure Appl. Algebra"},{"key":"726_CR26","doi-asserted-by":"crossref","unstructured":"Giusti, M., Heintz, J., Morais, J.E., Pardo, L.M.: When polynomial equation systems can be solved fast? In: Cohen, G., Giusti, M., Mora, T. (eds.) Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Proceedings AAECC-11. Lecture Notes in Comput. Sci., vol. 948, pp. 205\u2013231. Springer, Berlin (1995)","DOI":"10.1007\/3-540-60114-7_16"},{"key":"726_CR27","doi-asserted-by":"publisher","first-page":"1223","DOI":"10.1016\/S0764-4442(97)83558-6","volume":"325","author":"M Giusti","year":"1997","unstructured":"Giusti, M., Heintz, J., Morais, J.E., Pardo, L.M.: Le r\u00f4le des structures de donn\u00e9es dans les probl\u00e8mes d\u2019\u00e9limination. C. R. Math. Acad. Sci. Paris 325, 1223\u20131228 (1997)","journal-title":"C. R. Math. Acad. Sci. Paris"},{"key":"726_CR28","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1007\/BF01200407","volume":"3","author":"M Giusti","year":"1993","unstructured":"Giusti, M., Heintz, J., Sabia, J.: On the efficiency of effective Nullstellens\u00e4tze. Comput. Complex. 3, 56\u201395 (1993)","journal-title":"Comput. Complex."},{"issue":"1","key":"726_CR29","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1006\/jcom.2000.0571","volume":"17","author":"M Giusti","year":"2001","unstructured":"Giusti, M., Lecerf, G., Salvy, B.: A Gr\u00f6bner free alternative for polynomial system solving. J. Complex. 17(1), 154\u2013211 (2001)","journal-title":"J. Complex."},{"key":"726_CR30","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2189-8","volume-title":"Algebraic Geometry: A First Course. Grad. Texts in Math.,","author":"J Harris","year":"1992","unstructured":"Harris, J.: Algebraic Geometry: A First Course. Grad. Texts in Math., vol. 133. Springer, New York (1992)"},{"key":"726_CR31","doi-asserted-by":"crossref","unstructured":"Harvey, D., Hoeven, J.: Integer multiplication in time $$O(n\\log n)$$. Ann. Math. (2) 193(2), 563\u2013617 (2021)","DOI":"10.4007\/annals.2021.193.2.4"},{"issue":"3","key":"726_CR32","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/0304-3975(83)90002-6","volume":"24","author":"J Heintz","year":"1983","unstructured":"Heintz, J.: Definability and fast quantifier elimination in algebraically closed fields. Theor. Comput. Sci. 24(3), 239\u2013277 (1983)","journal-title":"Theor. Comput. Sci."},{"issue":"1","key":"726_CR33","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1006\/jcom.1999.0529","volume":"16","author":"J Heintz","year":"2000","unstructured":"Heintz, J., Krick, T., Puddu, S., Sabia, J., Waissbein, A.: Deformation techniques for efficient polynomial equation solving. J. Complex. 16(1), 70\u2013109 (2000)","journal-title":"J. Complex."},{"issue":"4","key":"726_CR34","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/s002000000046","volume":"11","author":"J Heintz","year":"2001","unstructured":"Heintz, J., Matera, G., Waissbein, A.: On the time-space complexity of geometric elimination procedures. Appl. Algebra Eng. Commun. Comput. 11(4), 239\u2013296 (2001)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"726_CR35","unstructured":"Heintz, J., Schnorr, C.P.: Testing polynomials which are easy to compute. In: International Symposium on Logic and Algorithmic, Zurich 1980. Monogr. Enseig. Math., vol. 30, pp. 237\u2013254 (1982)"},{"issue":"1","key":"726_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10208-020-09453-0","volume":"21","author":"J Hoeven","year":"2021","unstructured":"Hoeven, J., Lecerf, G.: On the complexity exponent of polynomial system solving. Found. Comput. Math. 21(1), 1\u201357 (2021)","journal-title":"Found. Comput. Math."},{"key":"726_CR37","volume-title":"Generic Local Structure of the Morphisms in Commutative Algebra. Lecture Notes in Math.","author":"B Iversen","year":"1973","unstructured":"Iversen, B.: Generic Local Structure of the Morphisms in Commutative Algebra. Lecture Notes in Math., vol. 310. Springer, New York (1973)"},{"key":"726_CR38","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10208-008-9024-2","volume":"9","author":"G Jeronimo","year":"2009","unstructured":"Jeronimo, G., Matera, G., Solern\u00f3, P., Waissbein, A.: Deformation techniques for sparse systems. Found. Comput. Math. 9, 1\u201350 (2009)","journal-title":"Found. Comput. Math."},{"key":"726_CR39","volume-title":"Th\u00e9or\u00e8mes de Bertini et Applications. Progr. Math","author":"JP Jouanolou","year":"1983","unstructured":"Jouanolou, J.P.: Th\u00e9or\u00e8mes de Bertini et Applications. Progr. Math. Birkh\u00e4user, Boston (1983)"},{"key":"726_CR40","first-page":"193","volume-title":"Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA\u201994. Progr. Math.,","author":"T Krick","year":"1996","unstructured":"Krick, T., Pardo, L.M.: A computational method for Diophantine approximation. In: Gonz\u00e1lez-Vega, L., Recio, T. (eds.) Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA\u201994. Progr. Math., vol. 143, pp. 193\u2013254. Birkh\u00e4user, Boston (1996)"},{"key":"726_CR41","first-page":"1","volume":"92","author":"L Kronecker","year":"1882","unstructured":"Kronecker, L.: Grundz\u00fcge einer arithmetischen Theorie der algebraischen Gr\u00f6ssen. J. Reine Angew. Math. 92, 1\u2013122 (1882)","journal-title":"J. Reine Angew. Math."},{"key":"726_CR42","volume-title":"Introduction to Commutative Algebra and Algebraic Geometry","author":"E Kunz","year":"1985","unstructured":"Kunz, E.: Introduction to Commutative Algebra and Algebraic Geometry. Birkh\u00e4user, Boston (1985)"},{"issue":"4","key":"726_CR43","doi-asserted-by":"publisher","first-page":"564","DOI":"10.1016\/S0885-064X(03)00031-1","volume":"19","author":"G Lecerf","year":"2003","unstructured":"Lecerf, G.: Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. Complex. 19(4), 564\u2013596 (2003)","journal-title":"J. Complex."},{"issue":"2","key":"726_CR44","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1007\/s00200-008-0062-4","volume":"19","author":"G Lecerf","year":"2008","unstructured":"Lecerf, G.: Fast separable factorization and applications. Appl. Algebra Eng. Commun. Comput. 19(2), 135\u2013160 (2008)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"726_CR45","volume-title":"Commutative Ring Theory","author":"H Matsumura","year":"1986","unstructured":"Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1986)"},{"key":"726_CR46","doi-asserted-by":"crossref","unstructured":"Moreno Maza, M., Rioboo, R.: Polynomial gcd computations over towers of algebraic extensions. In: Cohen, G., Giusti, M., Mora, T. (eds.) Proceedings of the 11th International Symposium on Applied Algebra, Algebraic Algorithms and Error\u2014Correcting Codes, AAECC\u201311. Lecture Notes in Comput. Sci., vol. 948, pp. 365\u2013382. Springer, Berlin (1995)","DOI":"10.1007\/3-540-60114-7_28"},{"key":"726_CR47","volume-title":"Algebraic Geometry I. Complex Projective Varieties, Classics Math.","author":"D Mumford","year":"1995","unstructured":"Mumford, D.: Algebraic Geometry I. Complex Projective Varieties, Classics Math., 2nd edn. Springer, Berlin (1995)","edition":"2"},{"key":"726_CR48","doi-asserted-by":"crossref","unstructured":"Pardo, L.M.: How lower and upper complexity bounds meet in elimination theory. In: Cohen, G., Giusti, M., Mora, T. (eds.) Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Proceedings of AAECC\u201311. Lecture Notes in Comput. Sci., vol. 948, pp. 33\u201369. Springer, Berlin (1995)","DOI":"10.1007\/3-540-60114-7_4"},{"issue":"2\u20133","key":"726_CR49","doi-asserted-by":"publisher","first-page":"593","DOI":"10.1016\/j.tcs.2004.01.009","volume":"315","author":"LM Pardo","year":"2004","unstructured":"Pardo, L.M., San Mart\u00edn, J.: Deformation techniques to solve generalized Pham systems. Theor. Comput. Sci. 315(2\u20133), 593\u2013625 (2004)","journal-title":"Theor. Comput. Sci."},{"key":"726_CR50","doi-asserted-by":"publisher","first-page":"101588","DOI":"10.1016\/j.jco.2021.101588","volume":"68","author":"LM Pardo","year":"2022","unstructured":"Pardo, L.M., Sebasti\u00e1n, D.: A promenade through correct test sequences. I: degree of constructible sets, B\u00e9zout\u2019s inequality and density. J. Complex. 68, 101588 (2022)","journal-title":"J. Complex."},{"key":"726_CR51","doi-asserted-by":"crossref","unstructured":"Pardo, L.M., Sebasti\u00e1n, D.: Erzeugunsgrad, VC-Dimension and neural networks with rational activation function. Appl. Algebra Eng. Commun. Comput. (2026)","DOI":"10.1007\/s00200-025-00723-4"},{"issue":"5","key":"726_CR52","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/s002000050114","volume":"9","author":"F Rouillier","year":"1997","unstructured":"Rouillier, F.: Solving zero-dimensional systems through rational univariate representation. Appl. Algebra Eng. Commun. Comput. 9(5), 433\u2013461 (1997)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"issue":"6","key":"726_CR53","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/BF01198015","volume":"6","author":"J Sabia","year":"1996","unstructured":"Sabia, J., Solern\u00f3, P.: Bounds for traces in complete intersections and degrees in the Nullstellensatz. Appl. Algebra Eng. Commun. Comput. 6(6), 353\u2013376 (1996)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"726_CR54","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s00200-002-0109-x","volume":"13","author":"E Schost","year":"2003","unstructured":"Schost, E.: Computing parametric geometric resolutions. Appl. Algebra Eng. Commun. Comput. 13, 349\u2013393 (2003)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"issue":"4","key":"726_CR55","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1145\/322217.322225","volume":"27","author":"JT Schwartz","year":"1980","unstructured":"Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701\u2013717 (1980)","journal-title":"J. ACM"},{"key":"726_CR56","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-57956-1","volume-title":"Basic Algebraic Geometry: Varieties in Projective Space","author":"IR Shafarevich","year":"1994","unstructured":"Shafarevich, I.R.: Basic Algebraic Geometry: Varieties in Projective Space. Springer, Berlin (1994)"},{"issue":"2","key":"726_CR57","first-page":"459","volume":"6","author":"M Shub","year":"1993","unstructured":"Shub, M., Smale, S.: Complexity of B\u00e9zout\u2019s theorem I: Geometric aspects. J. Am. Math. Soc. 6(2), 459\u2013501 (1993)","journal-title":"J. Am. Math. Soc."},{"key":"726_CR58","doi-asserted-by":"crossref","unstructured":"Sommese, A., Verschelde, J., Wampler, C.: Solving polynomial systems equation by equation. In: Dickenstein, A., Schreyer, F., Sommese, A. (eds.) Algorithms in Algebraic Geometry. IMA Vol. Math. Appl., vol. 146, pp. 133\u2013152. Springer (2008)","DOI":"10.1007\/978-0-387-75155-9_8"},{"key":"726_CR59","doi-asserted-by":"publisher","DOI":"10.1142\/5763","volume-title":"The Numerical Solution of Systems of Polynomials Arising in Engineering and Science","author":"A Sommese","year":"2005","unstructured":"Sommese, A., Wampler, C.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific, Singapore (2005)"},{"key":"726_CR60","unstructured":"Vogel, W.: Results on B\u00e9zout\u2019s Theorem. Tata Inst. Fundam. Res. Lect. Math., vol. 74. Tata Inst. Fund. Res, Bombay (1984)"},{"key":"726_CR61","doi-asserted-by":"crossref","unstructured":"Williams, V., Xu, Y., Xu, Z., Zhou, R.: New bounds for matrix multiplication: from alpha to omega. In: Proceedings of the 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, Virginia, January 7\u201310, 2024, pp. 3792\u20133835. SIAM, Philadelphia (2024)","DOI":"10.1137\/1.9781611977912.134"},{"key":"726_CR62","volume-title":"Commutative Algebra. Grad. Texts in Math.","author":"O Zariski","year":"1958","unstructured":"Zariski, O., Samuel, P.: Commutative Algebra. Grad. Texts in Math., vol. 38. Springer, New York (1958)"},{"key":"726_CR63","doi-asserted-by":"crossref","unstructured":"Zippel, R.: Probabilistic algorithms for sparse polynomials. In: EUROSAM \u201979: Proceedings of International Symposium on Symbolic and Algebraic Computation, Marseille 1979. Lecture Notes in Comput. Sci., vol. 72, pp. 216\u2013226. Springer, Berlin (1979)","DOI":"10.1007\/3-540-09519-5_73"}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-026-00726-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00200-026-00726-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-026-00726-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T09:08:07Z","timestamp":1776071287000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00200-026-00726-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3]]},"references-count":63,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["726"],"URL":"https:\/\/doi.org\/10.1007\/s00200-026-00726-9","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"value":"0938-1279","type":"print"},{"value":"1432-0622","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3]]},"assertion":[{"value":"20 May 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 April 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}