The Jacobian Conjecture and Inverse Degrees

Abstract

If k is a commutative ring and n is a positive integer we have the polynoinial algebra k ^[n] = k[x₁, ...,x_n] and the affine space EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaDa % aaleaacaWGRbaabaGaamOBaaaakiabg2da9iaadofacaWGWbGaamyz % aiaadogacaGGOaGaam4AamaaCaaaleqabaGaai4waiaad6gacaGGDb % aaaOGaaiykaaaa!42AC!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$A_k^n = Spec({k^{[n]}})$$ . A k-endomorphism F of EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaDa % aaleaacaWGRbaabaGaamOBaaaaaaa!38CA!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$A_k^n$$ will be identified with its sequence F = (F ₁,...,F _n) of coordinate functions EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa % aaleaacaWGPbaabeaakiabgIGiolaadUgadaahaaWcbeqaaiaacUfa % caWGUbGaaiyxaaaakiaaykW7caGGOaGaamyAaiabg2da9iaaigdaca % GGSaGaaiOlaiaac6cacaGGUaGaaiilaiaad6gacaGGPaaaaa!473D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${F_i} \in {k^{[n]}}\,(i = 1,...,n)$$ . Its Jacobian matrix is J(F) = (әF_i/әX_j). From the chain rule,

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaacI % cacaWGhbGaaiikaiaadAeacaGGPaGaaiykaiabg2da9iaadQeacaGG % OaGaam4raiaacMcacaGGOaGaamOraiaacMcacqGHflY1caWGkbGaai % ikaiaadAeacaGGPaaaaa!4667!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$J(G(F)) = J(G)(F) \cdot J(F)$$

we see that, if F is invertible then J(F) is invertible, i.e., det EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaacI % cacaWGgbGaaiykaiabgIGiolaacIcacaWGRbWaaWbaaSqabeaacaGG % BbGaamOBaiaac2faaaGccaGGPaWaaWbaaSqabeaacqGHxdaTaaaaaa!41E2!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$J(F) \in {({k^{[n]}})^ \times }$$ . (If k is reduced, i.e., has no nonzero nilpotent elements, then EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadU % gadaahaaWcbeqaaiaacUfacaWGUbGaaiyxaaaakiaacMcadaahaaWc % beqaaiabgEna0caakiabg2da9iaadUgadaahaaWcbeqaaiabgEna0c % aaaaa!41AF!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${({k^{[n]}})^ \times } = {k^ \times }$$ .) The Jacobian Conjecture asserts the converse when k is a field of characteristic zero: If J(F) is invertible then F is invertible.