Stronger Grace's theorem for complex polynomials using polar convexity
DOI:
https://doi.org/10.2140/mscand.2026.132.229Keywords:
polynomials, symmetrization, polar convexity, geometry of polynomials, polar derivative, apolarity, polar Gauss–Lucas theorem, Grace's theorem, Walsh's coincidence theorem, Walsh's representation theorem, Schur–Szegő composition theorem, De Bruijn's theorem, Rahman and Schmeisser's theoremAbstract
The classical theorem of J. H. Grace is a celebrated result in the geometric theory of polynomials. In this work, we present a strengthening of Grace's theorem using the notion of polar convexity. Moreover, this strengthening can be made even more precise assuming a simple nondegeneracy condition. It is well known that Grace's theorem has many equivalent formulations. Using polar convexity, we give the corresponding stronger versions of five equivalent restatements: Walsh's coincidence theorem, Walsh's representation theorem, the Schur–Szegő composition theorem, De Bruijn's theorem, and an equivalent version given by Rahman and Schmeisser. Our approach allowed us to detect a small imprecision in the original Schur–Szegő composition theorem.
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