.. include:: ../globals.def

.. _thesis-glossary:

========
Glossary
========

.. glossary::

    polynomial
        http://eom.springer.de/P/p073690.htm

    admissible ordering of monomials
        An ordering, $>$, on $\N^n$ is called an admissible ordering of monomials
        when it satisfies the following conditions:

        #. $>$ is a total ordering on $\N^n$
        #. $\alpha, \beta, \gamma \in \N^n$ and $\alpha > \beta$, then $\alpha + \gamma > \beta + \gamma$
        #. $\alpha \ge 0$ for all $\alpha \in \N^n$

    algebraic variety
        The set of solutions to a system of polynomial equations.

    characteristic zero
        A property of algebraic structures. An algebraic structure is of characteristic
        zero if it has infinite number of elements. Trivial examples are the ring of
        integers or the field of rational numbers. Contrary, an algebraic structure
        can be of positive characteristic, if it has finite number of elements. Then
        the characteristic tells how many elements are in the structure. In SymPy we
        can check if a domain has zero characteristic using :attr:`has_CharacteristicZero`.

    symmetric polynomial
        A polynomial $f \in \R\Xn$, where $\R$ is a ring, is called symmetric if $\sigma(f) = f$
        holds for every permutation $\sigma$ of the set $\{x_1, \ldots, x_n\}$. Every symmetric
        polynomial can be rewritten in terms of elementary symmetric polynomials utilizing method
        called symmetric reduction. This can can be accomplished in SymPy using :func:`symmetrize`
        function.

    elementary symmetric polynomial
        A polynomial $f \in \R\Xn$, where $\R$ is a ring, is called an elementary symmetric
        polynomial if $f$ is symmetric and $f$ is an element of the basis which generates all
        symmetric polynomials. The $k$--th elementary symmetric polynomial in $n$ variables,
        i.e. a polynomial of degree $k$, can be constructed by summing all different $k$--th
        degree monomials in $\{x_1, \ldots, x_n\}$. Elementary symmetric polynomials can be
        generated in SymPy using :func:`symmetric_poly` function.

    monic polynomial
        A polynomial with the leading coefficient equal to the identity element of the
        ground domain, or to zero if the polynomial itself is zero. In SymPy this can
        can be checked using `is_monic` property on an instance of :class:`Poly` class.

.. TODO: http://planetmath.org/encyclopedia/SymmetricPolynomial.html
.. TODO: http://planetmath.org/encyclopedia/ElementarySymmetricPolynomial.html