A Linear-algebraic Proof of Hilbert’s Ternary Quartic Theorem

Abstract

Hilbert’s ternary quartic theorem states that every nonnegative degree 4 homogeneous polynomial in three variables can be written as a sum of three squares of homogeneous quadratic polynomials. We give a linear-algebraic approach to Hilbert’s theorem by showing that a structured cone of positive semidefinite matrices is generated by rank 1 elements.

• MSC: Primary 11E20
• Secondary 15B48
• 90C22

Acknowledgments

The authors wish to thank the anonymous referees for their thoughtful comments which led to improvement in the presentation. They also thank Benjamin Grossmann for reading the article and providing helpful feedback. HJW is supported by Simons Foundation grant 355645.

Additional information

Notes on contributors

Anatolii Grinshpan

Anatolii Grinshpan received his Ph.D. from the University of California at Berkeley in 2001. He held appointments at the California Institute of Technology, the University of California at Berkeley, and Oklahoma State University. He is currently an associate teaching professor at Drexel University. His research interests include function theory and operator theory.

Hugo J. Woerdeman

Hugo J. Woerdeman received his Ph.D. from the Vrije Universiteit in Amsterdam, the Netherlands in 1989. In that same year he was appointed assistant professor at the College of William and Mary. During his tenure there he received a 1995 Alumni Fellowship Award for “Excellence in Teaching,” and he was awarded the title of Margaret L. Hamilton Professor of Mathematics. Since January 2005 he has been professor at Drexel University. Over the years, he has had long-term stays at the University of California, San Diego, George Washington University, École Nationale Supérieure de Techniques Avancées, Katholieke Universiteit Leuven, Université Catholique de Louvain, Princeton University, and the University of Waterloo. His research interests include matrix and operator theory and moment and factorization problems, often inspired by problems in systems theory, signal and image processing, and quantum information.