Birch's theorem

Birch's theorem

In mathematics, Birch's theorem, [B. J. Birch, "Homogeneous forms of odd degree in a large number of variables", Mathematika, 4, pages 102-105 (1957)] named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.

tatement of Birch's theorem

Let "K" be an algebraic number field, "k", "l" and "n" be natural numbers, "r"1,...,"r""k" be odd natural numbers, and "f"1,...,"f""k" be homogeneous polynomials with coefficients in "K" of degrees "r"1,...,"r""k" respectively in "n" variables, then there exists a number ψ("r"1,...,"r""k","l","K") such that:ngepsi(r_1,ldots,r_k,l,K)implies that there exists an "l"-dimensional vector subspace "V" of "K""n" such that:f_1(x)=ldots f_k(x)=0,quadforall xin V.

Remarks

The proof of the theorem is by induction over the maximal degree of the forms "f"1,...,"f""k". Essential to the proof is a special case, which can be proved by an application of the Hardy-Littlewood circle method, of the theorem which states that if "n" is sufficiently large and "r" is odd, then the equation:c_1x_1^r+ldots+c_nx_n^r=0,quad c_iinmathbb{Z}, i=1,ldots,nhas a solution in integers "x"1,...,"x""n", not all of which are 0.

The restriction to odd "r" is necessary, since even-degree forms, such as positive definite quadratic forms, may take the value 0 only at the origin.

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Birch and Swinnerton-Dyer conjecture — Millennium Prize Problems P versus NP problem Hodge conjecture Poincaré conjecture Riemann hypo …   Wikipedia

  • Bryan John Birch — F.R.S. (born 1931) is a British mathematician. His name has been given to the Birch and Swinnerton Dyer conjecture.As a doctoral student at the University of Cambridge, he was officially working under J. W. S. Cassels. More influenced by Harold… …   Wikipedia

  • Tunnell's theorem — In number theory, Tunnell s theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton Dyer conjecture, a full resolution. The congruent number problem asks which rational numbers can be the area of a… …   Wikipedia

  • Conjecture de Birch et Swinnerton-Dyer — Pour les articles homonymes, voir BSD (homonymie). En mathématiques, la conjecture de Birch (en) et Swinnerton Dyer prédit que pour toute courbe elliptique sur le corps des rationnels, l ordre d annulation en 1 de la fonc …   Wikipédia en Français

  • Bryan Birch — Bryan John Birch (* 25. September 1931 in Burton upon Trent) ist ein britischer Mathematiker, der sich mit Zahlentheorie beschäftigt. Er ist Professor an der Universität Oxford. Birch promovierte 1958 bei John Cassels in Cambridge (The geometry… …   Deutsch Wikipedia

  • Mordell–Weil theorem — In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K rational points of A is a finitely generated abelian group, called the Mordell Weil group. The case with A an elliptic curve… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • Hasse principle — In mathematics, Helmut Hasse s local global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each… …   Wikipedia

  • Hardy–Littlewood circle method — In mathematics, the Hardy–Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring s problem. Contents 1… …   Wikipedia

  • Glossary of arithmetic and Diophantine geometry — This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”