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  2. Rational point - Wikipedia

    en.wikipedia.org/wiki/Rational_point

    In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point.

  3. Rational root theorem - Wikipedia

    en.wikipedia.org/wiki/Rational_root_theorem

    The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial ( x – r ) can be factored out of the polynomial using polynomial long division , resulting in a polynomial of lower degree ...

  4. Rational function - Wikipedia

    en.wikipedia.org/wiki/Rational_function

    In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers ; they may be taken in any field K .

  5. Auxiliary function - Wikipedia

    en.wikipedia.org/wiki/Auxiliary_function

    For algebraic reasons this product is necessarily an integer, and using arguments relating to Wronskians it can be shown that it is non-zero, so its absolute value is an integer Ω ≥ 1. Using a version of the mean value theorem for matrices it is possible to get an analytic bound on Ω as well, and in fact using big-O notation we have

  6. Arithmetic geometry - Wikipedia

    en.wikipedia.org/wiki/Arithmetic_geometry

    The hyperelliptic curve defined by = (+) (+) has only finitely many rational points (such as the points (,) and (,)) by Faltings's theorem.. In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. [1]

  7. Rational number - Wikipedia

    en.wikipedia.org/wiki/Rational_number

    In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational ...

  8. Rational variety - Wikipedia

    en.wikipedia.org/wiki/Rational_variety

    This definition differs from that of path connectedness only by the nature of the path, but is very different, as the only algebraic curves which are rationally connected are the rational ones. Every rational variety, including the projective spaces, is rationally connected, but the converse is false. The class of the rationally connected ...

  9. Archimedean property - Wikipedia

    en.wikipedia.org/wiki/Archimedean_property

    By Ostrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some -adic absolute value. The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete.