Positivity Conditions for Hermitian Polynomials

Abstract We establish precise relationships among four distinct positivity conditions for bihomogeneous polynomials. We consider various one-parameter families $$r_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> of metrics on the line bundle $$U^{m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>U</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:math> over projective space $${\mathbb P}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> and give precise results about how these positivity conditions depend on t and m .