Examples

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User documentation

Let M denote a matrix. Let z denote an indeterminate in a polynomial ring.

 RingElem MinPoly(ConstMatrixView M, ConstRefRingElem z);

the minimal polynomial (in z) of M.

Let P denote a SparsePolyRing over Z/(p) or QQ. Let I denote an ideal in P. Let f denote an element of P. Let fbar denote an element of P/I. Let f denote an element of P. Let NumChecks denote a VerificationLevel.

• RingElem MinPoly(fbar, z) -- the minimal polynomial (in z) of fbar.

• RingElem MinPolyQuot(f, I, z, NumChecks) -- the minimal polynomial (in z) of f modulo I. Uses modular computation and MinPolyQuotDef. The modular computation is verified over NumChecks new primes. For NumChecks=0 there is full verification over QQ.
  RingElem MinPolyQuot(f, I, z) -- same as MinPolyQuot(f,I,z, 0)

    RingElem MinPolyQuotMat(ConstRefRingElem f, const ideal& I, ConstRefRingElem z);
    RingElem MinPolyQuotDef(ConstRefRingElem f, const ideal& I, ConstRefRingElem z);
    RingElem MinPolyQuotDefLin(ConstRefRingElem f, const ideal& I, ConstRefRingElem z);
    RingElem MinPolyQuotElim(ConstRefRingElem f, const ideal& I, ConstRefRingElem z);
    RingElem MinPolyMat(ConstRefRingElem fbar, ConstRefRingElem z);
    RingElem MinPolyDef(ConstRefRingElem fbar, ConstRefRingElem z);
    RingElem MinPolyElim(ConstRefRingElem fbar, ConstRefRingElem z);
  

  specific implementations (not modular: computation actually in QQ if coeff are in QQ)
  See article Abbott, Bigatti, Palezzato, Robbiano "Computing and Using Minimal Polynomials" ("https://arxiv.org/abs/1702.07262")

    matrix FrobeniusMat(const ideal& I);
    matrix FrobeniusMat(const ideal& I, const std::vector<PPMonoidElem>& QB2);
  
    std::vector<RingElem> ShapeLemma(const ideal& I);
  

Maintainer documentation

Main changes

2018

• August (v0.99600):
  • added documentation -