Adamus, Elżbieta; Bogdan, Paweł; Crespo, Teresa; Hajto, Zbigniew An effective study of polynomial maps. (English) Zbl 1371.14066 J. Algebra Appl. 16, No. 8, Article ID 1750141, 13 p. (2017). The extensively studied Jacobian conjecture asks whether every polynomial map \(F= (F_1, \dots, F_n): k^n \to k^n\) with \(\det DF\) a nonzero constant must have a polynomial inverse over fields \(k\) of characteristic zero. At the expense of adding more variables, the conjecture has been reduced to maps of the form \(F_i = x_i + H_i (x_1, \dots, x_n)\) where \(H_i\) is a cubic homogeneous map by work of H. Bass et al. [Bull. Am. Math. Soc., New Ser. 7, 287–330 (1982; Zbl 0539.13012)].For such maps in which \(d_i > 1\) is the least degree of a form in \(H_i\) and \(D_i = \deg H_i\), the authors give a new criterion for invertibility of \(F\). To state their result, let \(k[x]^n\) be the set of all polynomial maps \(F: k^n \to k^n\). Then \(\sigma_F (Q) = Q \circ F\) defines an endomorphism of \(k[x]^n\) and \(\Delta_F (Q)= \sigma_F (Q) - Q\) defines a \(\sigma_F\)-derivation on \(k[x]^n\). Now set \(P_k = \Delta_F^k (\text{Id})\) with components \(P_k = (P_k^1, \dots, P_k^n)\), The main result says that \(F\) is invertible if and only if for each \(1 \leq i \leq n\) there is a polynomial \(G_i\) of degree \(\leq D^{n-1}\) such that \[ R^i (x) = \sum_{j=0}^{m_i - 1} (-1)^j P_j^i (x) - G_i (x) \] satisfies the equation \(R^i (F) = (-1)^{m_i + 1} P_{m_i}^i (x)\), where \(m_i = \lfloor \frac{D^{n-1} - d_i}{d-1} + 1 \rfloor + 1\), in which case \(G = (G_1, \dots, G_n)\) is the inverse of \(F\). They also give a formula for \(G\) in terms of the \(P_k\). They use the software Maple to give some concrete examples for families of invertible \(F\) in terms of parameters and their inverses \(G\). Reviewer: Scott Nollet (Fort Worth) Cited in 1 ReviewCited in 2 Documents MSC: 14R15 Jacobian problem 14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Keywords:Jacobian conjecture; Pascal finite polynomial maps Citations:Zbl 0539.13012 Software:Maple PDFBibTeX XMLCite \textit{E. Adamus} et al., J. Algebra Appl. 16, No. 8, Article ID 1750141, 13 p. (2017; Zbl 1371.14066) Full Text: DOI References: [1] E. Adamus, P. Bogdan and Z. Hajto, An effective approach to Picard-Vessiot theory and the Jacobian Conjecture, arXiv:1506.01662 [math.AC], submitted. [2] Bass, H., Connell, E. and Wright, D., The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc.7 (1982) 287-330. · Zbl 0539.13012 [3] de Bondt, M., Quasi-translations and counterexamples to the homogeneous dependence problem, Proc. Amer. Math. Soc.134 (2006) 2849-2856. · Zbl 1107.14054 [4] M. de Bondt, Quasi-translations and singular Hessians, arXiv:1501.05168 [math.AG]. · Zbl 1107.14054 [5] Campbell, L. A., A condition for a polynomial map to be invertible, Math. Ann.205 (1973) 243-248. · Zbl 0245.13005 [6] Crespo, T. and Hajto, Z., Picard-Vessiot theory and the Jacobian problem, Israel J. Math.186 (2011) 401-406. · Zbl 1282.12003 [7] van den Essen, A., Polynomial Automorphisms and the Jacobian Conjecture, , Vol. 190 (Birkhäuser Verlag, 2000). · Zbl 0962.14037 [8] van den Essen, A., Seven lectures on polynomial automorphisms, Automorphisms of Affine Spaces (Curaçao, 1994), (Kluwer Academic Publishers, Dordrecht, 1995), pp. 3-39. · Zbl 0841.13006 [9] Furter, J.-Ph. and Maubach, S., Locally finite polynomial endomorphisms, J. Pure Appl. Algebra211 (2007) 445-458. · Zbl 1127.14054 [10] Gorni, G. and Zampieri, G., Yagzhev polynomial mappings: On the structure of the Taylor expansion of their local inverse, Ann. Polon. Math.64 (1996) 285-290. · Zbl 0868.12001 [11] Keller, O. H., Ganze Cremona-transformationen, Monatsh. Math. Phys.47 (1939) 299-306. · JFM 65.0713.02 [12] S. Maubach, Polynomial endomorphisms and kernels of derivations, Ph.D. thesis, Nijmegen University (2003), http://webdoc.ubn.kun.nl/mono/m/maubach_s/polyenank.pdf. [13] Rusek, K. and Winiarski, T., Polynomial automorphisms of \(\mathbb{C}^n\), Univ. Iagel. Acta Math.24 (1984) 143-149. · Zbl 0551.32020 [14] Wang, S., A Jacobian criterion for separability, J. Algebra65 (1980) 453-494. · Zbl 0471.13005 [15] Yagzhev, A. V., On a problem of O. H. Keller, Sibirsk. Mat. Zh.21 (1980) 747-754. · Zbl 0466.13009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.