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getHilbertSymbolReal -- computes the Hilbert symbol of two rational numbers over the real numbers

Description

The Hasse-Witt invariant of a diagonal form $\langle a_1,\ldots,a_n\rangle$ over a field $k$ is defined to be the product $\prod_{i<j} \left( a_i,a_j \right)_{\mathbb{R}}$ where $(-,-)_{\mathbb{R}}$ is the Hilbert symbol ([S73, Chapter III]) computed as follows:

$(a,b)_{\mathbb{R}} = \begin{cases} 1 & z^2 = ax^2 + by^2 \text{ has a nonzero solution in } {\mathbb{R}}^3 \\ -1 & \text{otherwise.} \end{cases}$

$(a,b)_{\mathbb{R}}$ will equal 1 unless both $a,\,b$ are negative.

Consider the example, that $z^2=-3x^2-2y^2/3$ does not admit a non-zero solution. Thus:

i1 : getHilbertSymbolReal(-3, -2/3) == -1

o1 = true

Computing Hasse-Witt invariants is a key step in classifying symmetric bilinear forms over the rational numbers, and in particular certifying their (an)isotropy.

Citations:

See also

Ways to use getHilbertSymbolReal:

  • getHilbertSymbolReal(QQ,QQ)
  • getHilbertSymbolReal(QQ,ZZ)
  • getHilbertSymbolReal(ZZ,QQ)
  • getHilbertSymbolReal(ZZ,ZZ)

For the programmer

The object getHilbertSymbolReal is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/HilbertSymbolsDoc.m2:74:0.