Algebraic curve in the unit ball in C² passing through the center, all whose boundary components are arbitrarily short

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We prove that curves indicated in the title exist. This results answers to a question posed by A.G.Vitushkin about 30 years ago. We also discuss the minimal number of boundary components of a curve in the unit ball passing through the center, under the condition that all these components are shorter than a given number. More precisely, we discuss the order of growth of the number of the components as their maximal length tends to zero.