@article{grosdos_heaton_kubjas_kuznetsova_scholten_sorea_2023, title={Exact solutions in log-concave maximum likelihood estimation}, volume={143}, ISSN={["1090-2074"]}, DOI={10.1016/j.aam.2022.102448}, abstractNote={We study probability density functions that are log-concave. Despite the space of all such densities being infinite-dimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many quantities, namely the function values, or heights, at the data points. We explore in what sense exact solutions to this problem are possible. First, we show that the heights given by the maximum likelihood estimate are generically transcendental. For a cell in one dimension, the maximum likelihood estimator is expressed in closed form using the generalized W-Lambert function. Even more, we show that finding the log-concave maximum likelihood estimate is equivalent to solving a collection of polynomial-exponential systems of a special form. Even in the case of two equations, very little is known about solutions to these systems. As an alternative, we use Smale's α-theory to refine approximate numerical solutions and to certify solutions to log-concave density estimation.}, journal={ADVANCES IN APPLIED MATHEMATICS}, author={Grosdos, Alexandros and Heaton, Alexander and Kubjas, Kaie and Kuznetsova, Olga and Scholten, Georgy and Sorea, Miruna-Stefana}, year={2023}, month={Feb} }
 @article{hong_kim_scholten_sendra_2017, title={Resultants over commutative idempotent semirings I: Algebraic aspect}, volume={79}, ISSN={["1095-855X"]}, url={http://dx.doi.org/10.1016/j.jsc.2016.02.009}, DOI={10.1016/j.jsc.2016.02.009}, abstractNote={The resultant theory plays a crucial role in computational algebra and algebraic geometry. The theory has two aspects: algebraic and geometric. In this paper, we focus on the algebraic aspect. One of the most important and well known algebraic properties of the resultant is that it is equal to the determinant of the Sylvester matrix. In 2008, Odagiri proved that a similar property holds over the tropical semiring if one replaces subtraction with addition. The tropical semiring belongs to a large family of algebraic structures called commutative idempotent semiring. In this paper, we prove that the same property (with subtraction replaced with addition) holds over an arbitrary commutative idempotent semiring.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Hong, Hoon and Kim, Yonggu and Scholten, Georgy and Sendra, J. Rafael}, year={2017}, pages={285–308} }