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Multivariate polynomial minimization and its application in signal processing. (English) Zbl 1023.90064
Summary: We make a conjecture that the number of isolated local minimum points of a \(2n\)-degree or \((2n+1)\)-degree \(r\)-variable polynomial is not greater than \(n^r\) when \(n\leqslant 2\). We show that this conjecture is the minimal estimate, and is true in several cases. In particular, we show that a cubic polynomial of \(r\) variables may have at most one local minimum point though it may have \(2^r\) critical points. We then study the global minimization problem of an even-degree multivariate polynomial whose leading order coefficient tensor is positive definite. We call such a multivariate polynomial a normal multivariate polynomial. By giving a one-variable polynomial majored below a normal multivariate polynomial, we show the existence of a global minimum of a normal multivariate polynomial, and give an upper bound of the norm of the global minimum and a lower bound of the global minimization value. We show that the quartic multivariate polynomial arising from broad-band antenna array signal processing, is a normal polynomial, and give a computable upper bound of the norm of the global minimum and a computable lower bound of the global minimization value of this normal quartic multivariate polynomial. We give some sufficient and necessary conditions for an even order tensor to be positive definite. Several challenging questions remain open.

MSC:
90C30 Nonlinear programming
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
26C10 Real polynomials: location of zeros
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
14Q99 Computational aspects in algebraic geometry
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