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On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers

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In this article we analyze the global diffeomorphism property of polynomial maps \(F:\mathbb {R}^n\rightarrow \mathbb {R}^n\) by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials \(\Vert F\Vert _2^2\). This allows us to identify a class of polynomial maps F for which their global diffeomorphism property on \(\mathbb {R}^n\) is equivalent to their Jacobian determinant \(\det JF\) vanishing nowhere on \(\mathbb {R}^n\). In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.

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Acknowledgements

The authors are grateful to Yu. Nesterov and V. Shikhman for pointing out the importance of the invariance of coercivity under linear transformations, and for other fruitful discussions on the subject of this article. The authors also wish to thank an anonymous referee for substantial remarks which significantly improved the quality of this manuscript.

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Authors and Affiliations

  1. Institute for Mathematics, Goethe-University, Frankfurt am Main, Germany

    Tomáš Bajbar

  2. Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Oliver Stein

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  1. Tomáš Bajbar
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Correspondence to Tomáš Bajbar.

A Appendix

A Appendix

1.1 A.1 Proof of Lemma 1

For any \(\bar{v}\in {{\mathrm{vert}}}(P+P)\) there exists a vector \(a\in \mathbb {R}^n{\setminus }\{0\}\) such that \(\bar{v}\) is the unique optimal point of the problem

$$\begin{aligned} \text {(LP1)}\quad \max _{v\in P+P}a^Tv. \end{aligned}$$

Let \(\bar{w}\in {{\mathrm{vert}}}(P)\) be an optimal point of the problem

$$\begin{aligned} \text {(LP2)}\quad \max _{w\in P}a^Tw. \end{aligned}$$

Since

$$\begin{aligned} \max _{v\in P+P}a^Tv= & {} \max _{(x,y)\in P\times P}a^T(x+y)=\max _{x\in P}a^Tx+\max _{y\in P}a^Ty\nonumber \\= & {} 2\max _{x\in P}a^Tx=2a^T\bar{w}=a^T2\bar{w} \end{aligned}$$
(A.1)

holds, the point \(2\bar{w}\in 2{{\mathrm{vert}}}(P)\) is an optimal point of (LP1). A a was chosen such that \(\bar{v}\) is the unique optimal point of (LP1), one obtains \(\bar{v}=2\bar{w}\) with \(\bar{w}\in {{\mathrm{vert}}}(P)\).

On the other hand, choose \(\bar{w}\in {{\mathrm{vert}}}(P)\) and put \(\bar{v}=2\bar{w}\). To show is \(\bar{v}\in {{\mathrm{vert}}}(P+P)\). Observe that there exists some \(a\in \mathbb {R}^n{\setminus }\{0\}\) such that \(\bar{w}\) is the unique optimal point of the problem (LP2). Using (A.1), the point \(\bar{v}\) is thus an optimal point of (LP1). Assume that \(\bar{v}\notin {{\mathrm{vert}}}(P+P)\) holds. Since (LP1) must possess a vertex solution, there exists an optimal point \(\bar{z}:=\bar{x}+\bar{y}\in P+P\) of (LP1) with \(\bar{v}\not =\bar{z}\). For the point \(\bar{u}:=\textstyle {\frac{1}{2}}(\bar{x}+\bar{y})\in P\) we obtain the identity

$$\begin{aligned} a^T\bar{u}= \frac{1}{2}a^T\left( \bar{x}+\bar{y}\right) =\frac{1}{2}a^T\bar{z}=a^T\bar{w}, \end{aligned}$$

where the last equation holds since both \(\bar{z}\) and \(\bar{v}=2 \bar{w}\) are optimal for (LP1). The point \(\bar{u}\in P\) is thus an optimal point of the problem (LP2), and the uniqueness of \(\bar{w}\) implies \(\bar{w}=\bar{u}\). This leads to the contradiction \(\bar{v}=\bar{z}\), and thus the assertion \(\bar{v}\in {{\mathrm{vert}}}(P+P)\) follows. \(\square \)

1.2 A.2 Proof of Lemma 5

Let \(S_n\) denote the symmetric group on n elements, let \({{\mathrm{sign}}}(\sigma )\) denote the permutation sign of \(\sigma \in S_n\), and for some arbitrarily given \(x\in \mathbb {R}^n\) let the entries of JF(x) be denoted by \(a_{ij}\), \(i,j\in I\). Then the Leibniz formula for determinants yields

$$\begin{aligned} \det JF(x)=\sum _{\sigma \in S_n}{{\mathrm{sign}}}(\sigma )\prod _{i\in I} a_{i,\sigma (i)} \end{aligned}$$

with

$$\begin{aligned} a_{i,\sigma (i)}=\frac{\partial }{\partial x_{\sigma (i)}}F_i(x) =\sum _{\alpha ^i\in A(F_i)}(F_i)_{\alpha ^i}\,\frac{\partial }{\partial x_{\sigma (i)}}x^{\alpha ^i} \end{aligned}$$

for all \(\sigma \in S_n\) and \(i\in I\). Interchanging multiplication and addition, and splitting the appearing products, further leads to

$$\begin{aligned} \prod _{i\in I} a_{i,\sigma (i)}= \sum _{(\alpha ^1,\ldots ,\alpha ^n)\in A(F_1)\times \dots \times A(F_n)}\left[ \prod _{i\in I}(F_i)_{\alpha ^i}\right] \cdot \left[ \prod _{i\in I}\frac{\partial }{\partial x_{\sigma (i)}}x^{\alpha ^i}\right] \end{aligned}$$

for all \(\sigma \in S_n\). In fact, in the above summation for any \(i\in I\) it is sufficient to choose \(\alpha ^i\in A(F_i)\) with \(\alpha ^i_{\sigma (i)}\ge 1\), since the existence of some \(j\in I\) with \(\alpha ^j_{\sigma (j)}=0\) means that the monomial \(x^{\alpha ^j}\) does not depend on the variable \(x_{\sigma (j)}\), resulting in

$$\begin{aligned} \frac{\partial }{\partial x_{\sigma (j)}}x^{\alpha ^j}= 0\quad \text {and}\quad \prod _{i\in I}\frac{\partial }{\partial x_{\sigma (i)}}x^{\alpha ^i}=0. \end{aligned}$$

This shows

$$\begin{aligned} \prod _{i\in I} a_{i,\sigma (i)}=\sum _{\begin{array}{c} \alpha ^i\in A(F_i),\,i\in I\\ \alpha ^i_{\sigma (i)}\ge 1,\,i\in I \end{array}}\left[ \prod _{i\in I}(F_i)_{\alpha ^i}\right] \cdot \left[ \prod _{i\in I}\frac{\partial }{\partial x_{\sigma (i)}}x^{\alpha ^i}\right] \end{aligned}$$

for all \(\sigma \in S_n\).

Next, for any \((\alpha ^1,\ldots ,\alpha ^n)\) in the above summation and any \(i\in I\) we have

$$\begin{aligned} \frac{\partial }{\partial x_{\sigma (i)}}x^{\alpha ^i}= \alpha ^i_{\sigma (i)}x^{\alpha ^i_{\sigma (i)}-1}_{\sigma _i}\prod _{j\ne i}x^{\alpha ^j_{\sigma (j)}}_{\sigma (j)} \end{aligned}$$

and, since \(\sigma \) is a permutation,

$$\begin{aligned} \prod _{i\in I}\frac{\partial }{\partial x_{\sigma (i)}}x^{\alpha ^i} =\left[ \prod _{i\in I}\alpha ^i_{\sigma (i)}\right] \cdot x^{\sum _{i\in I}\alpha ^i-\mathbbm {1}}. \end{aligned}$$

We arrive at

$$\begin{aligned} \prod _{i\in I} a_{i,\sigma (i)}= & {} \sum _{\begin{array}{c} \alpha ^i\in A(F_i),\,i\in I\\ \alpha ^i_{\sigma (i)}\ge 1,\,i\in I \end{array}}\left[ \prod _{i\in I}(F_i)_{\alpha ^i}\right] \cdot \left[ \prod _{i\in I}\alpha ^i_{\sigma (i)}\right] \cdot x^{\sum _{i\in I}\alpha ^i-\mathbbm {1}}\\= & {} \sum _{\begin{array}{c} \alpha ^i\in A(F_i),\,i\in I\\ \alpha ^i_{\sigma (i)}\ge 1,\,i\in I \end{array}}\left[ \prod _{i\in I}\alpha ^i_{\sigma (i)}\right] \cdot m(\alpha ^1,\ldots ,\alpha ^n,x) \end{aligned}$$

for all \(\sigma \in S_n\), where the monomial

$$\begin{aligned} m(\alpha ^1,\ldots ,\alpha ^n,x):=\left[ \prod _{i\in I}(F_i)_{\alpha ^i}\right] \cdot x^{\sum _{i\in I}\alpha ^i-\mathbbm {1}} \end{aligned}$$

does not depend on \(\sigma \). Hence, we may write

$$\begin{aligned} \det JF(x)= & {} \sum _{\sigma \in S_n}{{\mathrm{sign}}}(\sigma )\sum _{\begin{array}{c} \alpha ^i\in A(F_i),\,i\in I\\ \alpha ^i_{\sigma (i)}\ge 1,\,i\in I \end{array}}\left[ \prod _{i\in I}\alpha ^i_{\sigma (i)}\right] \cdot m(\alpha ^1,\ldots ,\alpha ^n,x)\\= & {} \sum _{\begin{array}{c} \alpha ^i\in A(F_i),\,i\in I \end{array}}\sum _{\begin{array}{c} \sigma \in S_n\\ \alpha ^i_{\sigma (i)}\ge 1,\,i\in I \end{array}}{{\mathrm{sign}}}(\sigma )\cdot \left[ \prod _{i\in I}\alpha ^i_{\sigma (i)}\right] \cdot m(\alpha ^1,\ldots ,\alpha ^n,x). \end{aligned}$$

In the latter outer summation it suffices to consider \(\alpha ^i\in A(F_i)\), \(i\in I\), with \(\sum _{i\in I}\alpha ^i\ge \mathbbm {1}\), since otherwise there would exist some \(j\in I\) with \(\alpha ^i_j=0\) for all \(i\in I\), resulting in \(\alpha ^{\sigma ^{-1}(j)}_j=0\) for any \(\sigma \in S_n\). However, then the inner summation would be taken over the empty set.

After introducing this restriction on the outer summation, we may drop the constraint \(\alpha ^i_{\sigma (i)}\ge 1\), \(i\in I\), in the inner summation since, for given \(\sigma \in S_n\), its violation leads to a vanishing product \(\prod _{i\in I}\alpha ^i_{\sigma (i)}\). Thus we have shown the assertion

$$\begin{aligned} \det JF(x)= & {} \sum _{\begin{array}{c} \alpha ^i\in A(F_i),\,i\in I\\ \sum _{i\in I}\alpha ^i\ge \mathbbm {1} \end{array}}\sum _{\sigma \in S_n}{{\mathrm{sign}}}(\sigma )\cdot \left[ \prod _{i\in I}\alpha ^i_{\sigma (i)}\right] \cdot m(\alpha ^1,\ldots ,\alpha ^n,x)\\= & {} \sum _{\begin{array}{c} \alpha ^i\in A(F_i),\,i\in I\\ \sum _{i\in I}\alpha ^i\ge \mathbbm {1} \end{array}}\det (\alpha ^1,\ldots ,\alpha ^n)\cdot m(\alpha ^1,\ldots ,\alpha ^n,x), \end{aligned}$$

where the final identity is due to the Leibniz formula for determinants. \(\square \)

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Bajbar, T., Stein, O. On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers. Math. Z. 288, 915–933 (2018). https://doi.org/10.1007/s00209-017-1920-1

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