It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a weighted sum of finitely many squares instead of a sum of two squares, then one can choose all coefficients in the representation to lie in the field generated by the coefficients of the polynomial. In particular, this allows an effect...

The problem of finding a piecewise straight-line path, with a constant number of line segments, in a two-dimensional domain is studied in the Turing machine-based computational model and in the discrete complexity theory. It is proved that, for polynomial-time recognizable domains associated with polynomial-time computable distance functions, the complexity of this problem is equivalent to a di...

In this paper, we study computability and complexity of real functions. We extend these notions, already defined for functions over closed intervals or over the real line to functions over particular real open sets and give some results and characterizations, especially for polynomial time computable functions. Our representation of real numbers as sequences of rational numbers allows us to imp...

The class UP of ‘ultimate polynomial time’ problems over C is introduced; it contains the class P of polynomial time problems over C. The τ -Conjecture for polynomials implies that UP does not contain the class of non-deterministic polynomial time problems definable without constants over C. This latest statement implies that P 6 = NP over C. A notion of ‘ultimate complexity’ of a problem is su...

We present several generalisations of the GamesChan algorithm. For a fixed monic irreducible polynomial f we consider the sequences s that have as characteristic polynomial a power of f . We propose an algorithm for computing the linear complexity of s given a full (not necessarily minimal) period of s. We give versions of the algorithm for fields of characteristic 2 and for arbitrary finite ch...

where equivalent under polynomial bounds. In other words, the complexity classes defined by parallel polynomial time, alternating polynomial time, and polynomial space are actually the same class. In contrast with the above, it was soon remarked that in the theory of complexity over the reals developed by Blum, Shub, and Smale [2], every decidable set can be decided using constant workspace [8]...

We continue the study of counting complexity begun in [13, 14, 15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a f...

We present several generalisations of the GamesChan algorithm. For a fixed monic irreducible polynomial f we consider the sequences s that have as characteristic polynomial a power of f . We propose an algorithm for computing the linear complexity of s given a full (not necessarily minimal) period of s. We give versions of the algorithm for fields of characteristic 2 and for arbitrary finite ch...

We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a p-adic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m=2 (and no classical randomized polynomial time algorithm is known), (⋆) i...

Every Boolean function is uniquely defined by a polynomial modulo 2. The degree of a Boolean function is the degree of its defining polynomial. In cryptography, the Boolean functions of fixed degree played important role, for example, 1 or 2 degrees. Therefore, in finding algorithms that recognize properties of Boolean functions polynomials by their values vectors, it makes sense consider only ...