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The question about uniqueness was flawed. I corrected it. I added constraints on the domain of $ g(X) $
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Schemer1
Schemer1
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If $ \vec{u} $ (respectively $ \vec{v} $) is an element of $ \mathbb{C}^{n} $ with components $ u_{i} $ (respectively $ v_{i} $) and $ \langle \vec{u},\vec{v} \rangle $ is the Hermitian inner product $ \sum_{i=1}^{n} u_{i} \overline{v_{i}} $, then does anyone know a form for solutions of the PDE $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle = -\frac{\partial g(X)}{\partial x_{1}} $ for a polynomial function $ g(X): \mathbb{C}^{n} \to \mathbb{C} $? Is such a solution unique up to addition of a constant in $ \mathbb{C} $? For any such solution does $ \frac{\partial g(X)}{\partial x_{1}} $ equal zero?

If $ \vec{u} $ (respectively $ \vec{v} $) is an element of $ \mathbb{C}^{n} $ with components $ u_{i} $ (respectively $ v_{i} $) and $ \langle \vec{u},\vec{v} \rangle $ is the Hermitian inner product $ \sum_{i=1}^{n} u_{i} \overline{v_{i}} $, then does anyone know a form for solutions of the PDE $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle = -\frac{\partial g(X)}{\partial x_{1}} $? Is such a solution unique? For any such solution does $ \frac{\partial g(X)}{\partial x_{1}} $ equal zero?

If $ \vec{u} $ (respectively $ \vec{v} $) is an element of $ \mathbb{C}^{n} $ with components $ u_{i} $ (respectively $ v_{i} $) and $ \langle \vec{u},\vec{v} \rangle $ is the Hermitian inner product $ \sum_{i=1}^{n} u_{i} \overline{v_{i}} $, then does anyone know a form for solutions of the PDE $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle = -\frac{\partial g(X)}{\partial x_{1}} $ for a polynomial function $ g(X): \mathbb{C}^{n} \to \mathbb{C} $? Is such a solution unique up to addition of a constant in $ \mathbb{C} $? For any such solution does $ \frac{\partial g(X)}{\partial x_{1}} $ equal zero?

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Schemer1
Schemer1
  • 912
  • 3
  • 12

Solution to the differential equation $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle =-\frac{\partial g(X)}{\partial x_{1}} $?

If $ \vec{u} $ (respectively $ \vec{v} $) is an element of $ \mathbb{C}^{n} $ with components $ u_{i} $ (respectively $ v_{i} $) and $ \langle \vec{u},\vec{v} \rangle $ is the Hermitian inner product $ \sum_{i=1}^{n} u_{i} \overline{v_{i}} $, then does anyone know a form for solutions of the PDE $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle = -\frac{\partial g(X)}{\partial x_{1}} $? Is such a solution unique? For any such solution does $ \frac{\partial g(X)}{\partial x_{1}} $ equal zero?