Addition of Chow ring ideal and Chow ring classes #38281
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| r""" | ||
| Chow rings of matroids | ||
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| AUTHORS: | ||
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| - Shriya M | ||
| """ | ||
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| from sage.matroids.chow_ring_ideal import ChowRingIdeal_nonaug, AugmentedChowRingIdeal_fy, AugmentedChowRingIdeal_atom_free | ||
| from sage.rings.quotient_ring import QuotientRing_generic | ||
| from sage.categories.graded_algebras_with_basis import GradedAlgebrasWithBasis | ||
| from sage.categories.commutative_rings import CommutativeRings | ||
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| class ChowRing(QuotientRing_generic): | ||
| r""" | ||
| The Chow ring of a matroid. | ||
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| The *Chow ring of the matroid* `M` is defined as the quotient ring | ||
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| .. MATH:: | ||
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| A^*(M)_R := R[x_{F_1}, \ldots, x_{F_k}] / (I_M + J_M), | ||
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| where `(I_M + J_M)` is the :class:`Chow ring ideal | ||
| <sage.matroids.chow_ring_ideal.ChowRingIdeal_nonaug>` of matroid `M`. | ||
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| The *augmented Chow ring of matroid* `M` has two different presentations | ||
| as quotient rings: | ||
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| The *Feitchner-Yuzvinsky presentation* is the quotient ring | ||
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| .. MATH:: | ||
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| A(M)_R := R[y_{e_1}, \ldots, y_{e_n}, x_{F_1}, \ldots, x_{F_k}] / I_{FY}(M), | ||
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| where `I_{FY}(M)` is the :class:`Feitchner-Yuzvinsky augmented Chow ring | ||
| ideal <sage.matroids.chow_ring_ideal.AugmentedChowRingIdeal_fy>` | ||
| of matroid `M`. | ||
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| The *atom-free presentation* is the quotient ring | ||
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| .. MATH:: | ||
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| A(M)_R := R[x_{F_1}, \ldots, x_{F_k}] / I_{af}(M), | ||
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| where `I_{af}(M)` is the :class:`atom-free augmented Chow ring ideal | ||
| <sage.matroids.chow_ring_ideal.AugmentedChowRingIdeal_atom_free>` | ||
| of matroid `M`. | ||
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| .. SEEALSO:: | ||
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| :mod:`sage.matroids.chow_ring_ideal` | ||
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| INPUT: | ||
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| - ``M`` -- matroid | ||
| - ``R`` -- commutative ring | ||
| - ``augmented`` -- boolean; when ``True``, this is the augmented | ||
| Chow ring and if ``False``, this is the non-augmented Chow ring | ||
| - ``presentation`` -- string (default: ``None``); one of the following | ||
| (ignored if ``augmented=False``) | ||
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| * ``"fy"`` - the Feitchner-Yuzvinsky presentation | ||
| * ``"atom-free"`` - the atom-free presentation | ||
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| REFERENCES: | ||
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| - [FY2004]_ | ||
| - [AHK2015]_ | ||
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| EXAMPLES:: | ||
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| sage: M1 = matroids.catalog.P8pp() | ||
| sage: ch = M1.chow_ring(QQ, False) | ||
| sage: ch | ||
| Chow ring of P8'': Matroid of rank 4 on 8 elements with 8 nonspanning circuits | ||
| over Rational Field | ||
| """ | ||
| def __init__(self, R, M, augmented, presentation=None): | ||
| r""" | ||
| Initialize ``self``. | ||
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| EXAMPLES:: | ||
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| sage: ch = matroids.Wheel(3).chow_ring(QQ, False) | ||
| sage: TestSuite(ch).run() | ||
| """ | ||
| self._matroid = M | ||
| self._augmented = augmented | ||
| self._presentation = presentation | ||
| if augmented is True: | ||
| if presentation == 'fy': | ||
| self._ideal = AugmentedChowRingIdeal_fy(M, R) | ||
| elif presentation == 'atom-free': | ||
| self._ideal = AugmentedChowRingIdeal_atom_free(M, R) | ||
| else: | ||
| self._ideal = ChowRingIdeal_nonaug(M, R) | ||
| C = CommutativeRings().Quotients() & GradedAlgebrasWithBasis(R).FiniteDimensional() | ||
| QuotientRing_generic.__init__(self, R=self._ideal.ring(), | ||
| I=self._ideal, | ||
| names=self._ideal.ring().variable_names(), | ||
| category=C) | ||
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| def _repr_(self): | ||
| r""" | ||
| EXAMPLES:: | ||
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| sage: M1 = matroids.catalog.Fano() | ||
| sage: ch = M1.chow_ring(QQ, False) | ||
| sage: ch | ||
| Chow ring of Fano: Binary matroid of rank 3 on 7 elements, type (3, 0) | ||
| over Rational Field | ||
| """ | ||
| output = "Chow ring of {}".format(self._matroid) | ||
| if self._augmented is True: | ||
| output = "Augmented " + output | ||
| if self._presentation == 'fy': | ||
| output += " in Feitchner-Yuzvinsky presentation" | ||
| elif self._presentation == 'atom-free': | ||
| output += " in atom-free presentation" | ||
| return output + " over " + repr(self.base_ring()) | ||
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| def _latex_(self): | ||
| r""" | ||
| Return the LaTeX output of the polynomial ring and Chow ring ideal. | ||
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| EXAMPLES:: | ||
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| sage: M1 = matroids.Uniform(2,5) | ||
| sage: ch = M1.chow_ring(QQ, False) | ||
| sage: ch._latex_() | ||
| 'A(\\begin{array}{l}\n\\text{\\texttt{U(2,{ }5):{ }Matroid{ }of{ }rank{ }2{ }on{ }5{ }elements{ }with{ }circuit{-}closures}}\\\\\n\\text{\\texttt{{\\char`\\{}2:{ }{\\char`\\{}{\\char`\\{}0,{ }1,{ }2,{ }3,{ }4{\\char`\\}}{\\char`\\}}{\\char`\\}}}}\n\\end{array})_{\\Bold{Q}}' | ||
| """ | ||
| from sage.misc.latex import latex | ||
| base = "A({})_{{{}}}" | ||
| if self._augmented: | ||
| base += "^*" | ||
| return base.format(latex(self._matroid), latex(self.base_ring())) | ||
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| def matroid(self): | ||
| r""" | ||
| Return the matroid of ``self``. | ||
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| EXAMPLES:: | ||
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| sage: ch = matroids.Uniform(3,6).chow_ring(QQ, True, 'fy') | ||
| sage: ch.matroid() | ||
| U(3, 6): Matroid of rank 3 on 6 elements with circuit-closures | ||
| {3: {{0, 1, 2, 3, 4, 5}}} | ||
| """ | ||
| M = self._matroid | ||
| return M | ||
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| def _coerce_map_from_base_ring(self): | ||
| r""" | ||
| Disable the coercion from the base ring from the category. | ||
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| TESTS:: | ||
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| sage: ch = matroids.Wheel(3).chow_ring(QQ, False) | ||
| sage: ch._coerce_map_from_base_ring() is None | ||
| True | ||
| """ | ||
| return None # don't need anything special | ||
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| def basis(self): | ||
| r""" | ||
| Return the monomial basis of the given Chow ring. | ||
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| EXAMPLES:: | ||
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| sage: ch = matroids.Uniform(3, 6).chow_ring(QQ, True, 'fy') | ||
| sage: ch.basis() | ||
| Family (1, B1, B1*B012345, B0, B0*B012345, B01, B01^2, B2, | ||
| B2*B012345, B02, B02^2, B12, B12^2, B3, B3*B012345, B03, B03^2, | ||
| B13, B13^2, B23, B23^2, B4, B4*B012345, B04, B04^2, B14, B14^2, | ||
| B24, B24^2, B34, B34^2, B5, B5*B012345, B05, B05^2, B15, B15^2, | ||
| B25, B25^2, B35, B35^2, B45, B45^2, B012345, B012345^2, B012345^3) | ||
| sage: set(ch.defining_ideal().normal_basis()) == set(ch.basis()) | ||
| True | ||
| sage: ch = matroids.catalog.Fano().chow_ring(QQ, False) | ||
| sage: ch.basis() | ||
| Family (1, Abcd, Aace, Aabf, Adef, Aadg, Abeg, Acfg, Aabcdefg, | ||
| Aabcdefg^2) | ||
| sage: set(ch.defining_ideal().normal_basis()) == set(ch.basis()) | ||
| True | ||
| sage: ch = matroids.Wheel(3).chow_ring(QQ, True, 'atom-free') | ||
| sage: ch.basis() | ||
| Family (1, A0, A0*A012345, A2, A2*A012345, A3, A3*A012345, A23, | ||
| A23^2, A1, A1*A012345, A013, A013^2, A4, A4*A012345, A04, A04^2, | ||
| A124, A124^2, A5, A5*A012345, A025, A025^2, A15, A15^2, A345, | ||
| A345^2, A012345, A012345^2, A012345^3) | ||
| sage: set(ch.defining_ideal().normal_basis()) == set(ch.basis()) | ||
| True | ||
| """ | ||
| from sage.sets.family import Family | ||
| monomial_basis = self._ideal.normal_basis() | ||
| return Family([self.element_class(self, mon, reduce=False) for mon in monomial_basis]) | ||
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| class Element(QuotientRing_generic.Element): | ||
| def to_vector(self, order=None): | ||
| r""" | ||
| Return ``self`` as a (dense) free module vector. | ||
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| EXAMPLES:: | ||
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| sage: ch = matroids.Uniform(3, 6).chow_ring(QQ, False) | ||
| sage: v = ch.an_element(); v | ||
| -A01 - A02 - A03 - A04 - A05 - A012345 | ||
| sage: v.to_vector() | ||
| (0, -1, -1, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0) | ||
| """ | ||
| P = self.parent() | ||
| B = P.basis() | ||
| FM = P._dense_free_module() | ||
| f = self.lift() | ||
| return FM([f.monomial_coefficient(b.lift()) for b in B]) | ||
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| _vector_ = to_vector | ||
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| def monomial_coefficients(self, copy=None): | ||
| r""" | ||
| Return the monomial coefficients of ``self``. | ||
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| EXAMPLES:: | ||
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| sage: ch = matroids.catalog.NonFano().chow_ring(QQ, True, 'atom-free') | ||
| sage: v = ch.an_element(); v | ||
| Aa | ||
| sage: v.monomial_coefficients() | ||
| {0: 0, 1: 1, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0, | ||
| 10: 0, 11: 0, 12: 0, 13: 0, 14: 0, 15: 0, 16: 0, 17: 0, | ||
| 18: 0, 19: 0, 20: 0, 21: 0, 22: 0, 23: 0, 24: 0, 25: 0, | ||
| 26: 0, 27: 0, 28: 0, 29: 0, 30: 0, 31: 0, 32: 0, 33: 0, | ||
| 34: 0, 35: 0} | ||
| """ | ||
| B = self.parent().basis() | ||
| f = self.lift() | ||
| return {i: f.monomial_coefficient(b.lift()) for i, b in enumerate(B)} | ||
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| def degree(self): | ||
| r""" | ||
| Return the degree of ``self``. | ||
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| EXAMPLES:: | ||
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| sage: ch = matroids.Uniform(3, 6).chow_ring(QQ, False) | ||
| sage: for b in ch.basis(): | ||
| ....: print(b, b.degree()) | ||
| 1 0 | ||
| A01 1 | ||
| A02 1 | ||
| A12 1 | ||
| A03 1 | ||
| A13 1 | ||
| A23 1 | ||
| A04 1 | ||
| A14 1 | ||
| A24 1 | ||
| A34 1 | ||
| A05 1 | ||
| A15 1 | ||
| A25 1 | ||
| A35 1 | ||
| A45 1 | ||
| A012345 1 | ||
| A012345^2 2 | ||
| sage: v = sum(ch.basis()) | ||
| sage: v.degree() | ||
| 2 | ||
| """ | ||
| return self.lift().degree() | ||
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| def homogeneous_degree(self): | ||
| r""" | ||
| Return the (homogeneous) degree of ``self`` if homogeneous | ||
| otherwise raise an error. | ||
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| EXAMPLES:: | ||
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| sage: ch = matroids.catalog.Fano().chow_ring(QQ, True, 'fy') | ||
| sage: for b in ch.basis(): | ||
| ....: print(b, b.homogeneous_degree()) | ||
| 1 0 | ||
| Ba 1 | ||
| Ba*Babcdefg 2 | ||
| Bb 1 | ||
| Bb*Babcdefg 2 | ||
| Bc 1 | ||
| Bc*Babcdefg 2 | ||
| Bd 1 | ||
| Bd*Babcdefg 2 | ||
| Bbcd 1 | ||
| Bbcd^2 2 | ||
| Be 1 | ||
| Be*Babcdefg 2 | ||
| Bace 1 | ||
| Bace^2 2 | ||
| Bf 1 | ||
| Bf*Babcdefg 2 | ||
| Babf 1 | ||
| Babf^2 2 | ||
| Bdef 1 | ||
| Bdef^2 2 | ||
| Bg 1 | ||
| Bg*Babcdefg 2 | ||
| Badg 1 | ||
| Badg^2 2 | ||
| Bbeg 1 | ||
| Bbeg^2 2 | ||
| Bcfg 1 | ||
| Bcfg^2 2 | ||
| Babcdefg 1 | ||
| Babcdefg^2 2 | ||
| Babcdefg^3 3 | ||
| sage: v = sum(ch.basis()); v | ||
| Babcdefg^3 + Babf^2 + Bace^2 + Badg^2 + Bbcd^2 + Bbeg^2 + | ||
| Bcfg^2 + Bdef^2 + Ba*Babcdefg + Bb*Babcdefg + Bc*Babcdefg + | ||
| Bd*Babcdefg + Be*Babcdefg + Bf*Babcdefg + Bg*Babcdefg + | ||
| Babcdefg^2 + Ba + Bb + Bc + Bd + Be + Bf + Bg + Babf + Bace + | ||
| Badg + Bbcd + Bbeg + Bcfg + Bdef + Babcdefg + 1 | ||
| sage: v.homogeneous_degree() | ||
| Traceback (most recent call last): | ||
| ... | ||
| ValueError: element is not homogeneous | ||
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| TESTS:: | ||
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| sage: ch = matroids.Wheel(3).chow_ring(QQ, True, 'atom-free') | ||
| sage: ch.zero().homogeneous_degree() | ||
| Traceback (most recent call last): | ||
| ... | ||
| ValueError: the zero element does not have a well-defined degree | ||
| """ | ||
| if not self: | ||
| raise ValueError("the zero element does not have a well-defined degree") | ||
| f = self.lift() | ||
| if not f.is_homogeneous(): | ||
| raise ValueError("element is not homogeneous") | ||
| return f.degree() | ||
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