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Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers ( Irrationalität und Transzendenz bestimmter Zahlen ).
Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6, [g] 9, 11, 12, 15, 21, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems.
In 1888, Hilbert showed that every non-negative homogeneous polynomial in n variables and degree 2d can be represented as sum of squares of other polynomials if and only if either (a) n = 2 or (b) 2d = 2 or (c) n = 3 and 2d = 4. [2] Hilbert's proof did not exhibit any explicit counterexample: only in 1967 the first explicit counterexample was ...
Pages in category "Hilbert's problems" The following 35 pages are in this category, out of 35 total. This list may not reflect recent changes. ...
Part of the seventh of Hilbert's twenty-three problems posed in 1900 was to prove, or find a counterexample to, the claim that a b is always transcendental for algebraic a ≠ 0, 1 and irrational algebraic b. In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2 √ 2.
Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether a solution exists for all 7th-degree equations using algebraic (variant: continuous ) functions of two arguments .
A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. [223] This list has achieved great celebrity among mathematicians, [224] and at least thirteen of the problems (depending how some are interpreted) have been solved. [223]
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite).