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Register a new userOre Polynomials in Sage
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Added by
Manuel Kauers
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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We present a Sage implementation of Ore algebras. The main features for the most common instances include basic arithmetic and actions; gcrd and lclm; D-finite closure properties; natural transformations between related algebras; guessing; desingularization; solvers for polynomials, rational functions and (generalized) power series. This paper is a tutorial on how to use the package.
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Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.
Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the $(r,d)$-plane such that for all points $(r,d)$ above this curve, there exists a left multiple of order $r$ and degree $d$ of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples.
Given a black box function to evaluate an unknown rational polynomial f in Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t, the shift s (a rational), the exponents 0 <= e1 < e2 < ... < et, and the coefficients c1,...,ct in Q{0} such that f(x) = c1(x-s)^e1+c2(x-s)^e2+...+ct(x-s)^et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, and in particular is logarithmic in deg(f). Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.
The aim of the paper is to produce new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form $F(x,y)=f_1(x)f_2(y)-f_2(x)f_1(y)$, then $F(x,y)+r$ is always irreducible for any constant $r$ different from zero. We also provide the biggest known family of HIP polynomials in several variables. These are polynomials $p(x_1,ldots,x_n) in K[x_1,ldots,x_n]$ over a zero characteristic field $K$ such that $p(h_1(x_1),ldots,h_n(x_n))$ is irreducible over $K$ for every $n$-tuple $h_1(x_1),ldots,h_n(x_n)$ of non constant one variable polynomials over $K$. The results can also be applied to fields of positive characteristic, with some modifications.
The texttt{StronglyStableIdeals} package for textit{Macaulay2} provides a method to compute all saturated strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. A description of the main method and auxiliary tools is given.
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