New families of nonnegative biquadratic forms that have 8, 9 or 10 real zeros in P 2 × P 2 are constructed. These are the first examples with 8, 9 or 10 real zeros. It is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros; our examples show that the upper bound is obtained. Such biquadratic forms define positive linear maps that are not completely positive. Our constructions are explicit, and moreover we are able to determine which of the examples are extremal.

Let N (d, n) be the variety of all d-tuples of commuting nilpotent n × n matrices. It is well-known that N (d, n) is irreducible if d = 2, if n ≤ 3 or if d = 3 and n = 4. On the other hand N (3, n) is known to be reducible for n ≥ 13. We study in this paper the reducibility of N (d, n) for various values of d and n. In particular, we prove that N (d, n) is reducible for all d, n ≥ 4. In the case d = 3, we show that it is irreducible for n ≤ 6.

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A * -linear map Φ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations I n ⊗ Φ are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.Date: February 22, 2018. 2010 Mathematics Subject Classification. 13J30, 46L07, 52A40 (Primary); 47L25, 81P45, 90C22 (Secondary).

Gerstenhaber's theorem states that the dimension of the unital algebra generated by two commuting n × n matrices is at most n. We study the analog of this question for positive matrices with a positive commutator. We show that the dimension of the unital algebra generated by the matrices is at most n(n+1) 2 and that this bound can be attained. We also consider the corresponding question if one of the matrices is a permutation or a companion matrix or both of them are idempotents. In these cases, the upper bound for the dimension can be reduced significantly. In particular, the unital algebra generated by two semi-commuting positive idempotent matrices is at most 9-dimensional. This upper bound can be attained.Math. Subj. Classification (2010): 15A27, 15B48, 47B47.