Implement preimage computation for Homomorphism from Quotient Ring to Finite Field of the same characteristic #39709

rasti37 · 2025-03-14T17:10:34Z

Technical Details

This PR introduces a new function into the file src/sage/rings/morphism.pyx, named _preimage_from_linear_dependence.
More specifically, it computes the preimage of an element for a homomorphism $f$ from a univariate polynomial quotient ring to a finite extension field extension. When the codomain's field degree is the same as the domain's base ring degree, and the domain is constructed from an irreducible polynomial, $f$ is guaranteed to have a preimage for any element and _preimage_from_linear_dependence returns this preimage.

Fixes #39690.

The way this function is currently used, it fixes the NotImplementedError raised by:

f.inverse
f.inverse_image and
f._inverse_image_element

How it was fixed?

Internally, _inverse_image_element calls _graph_ideal which is the cause of the NotImplementedError due to the check A.base_ring() != B.base_ring(). Now, before calling _graph_ideal, the new method _preimage_from_linear_dependence is conditionally called and returns early, thereby avoiding the error.

I have not included any documentation because it's supposed to be used as an internal function. Let me know if it is needed.

📝 Checklist

The title is concise and informative.
The description explains in detail what this PR is about.
I have linked a relevant issue or discussion.
I have created tests covering the changes.
I have updated the documentation and checked the documentation preview.

⌛ Dependencies

Fixing the ruff linting #39699 : Probably due to a recent version change in the ruff linter, the ruff linting check failed. This PR fixes these errors so I cherry-picked the author's commit.

…inting based on PR:39699

grhkm21

I will look at it more detailedly later once my Sage rebuilds.
By the way, please avoid touching random styling issues in the future, I know it annoys you (and me) but better do it in a separate PR (and we should have more of those!)

grhkm21 · 2025-03-19T23:17:18Z

src/sage/rings/morphism.pyx

+            sage: f._preimage_from_linear_dependence(z)
+            Traceback (most recent call last):
+            ...
+            ValueError: The cardinalities of the domain (=1024) and codomain (=64) should be equal.


Suggested change

ValueError: The cardinalities of the domain (=1024) and codomain (=64) should be equal.

ValueError: the cardinalities of the domain (=1024) and codomain (=64) should be equal

Done! (Check new commit)

grhkm21 · 2025-03-19T23:20:16Z

src/sage/rings/morphism.pyx

+        D = self.domain()
+        C = self.codomain()
+        if (d_card := D.cardinality()) != (c_card := C.cardinality()):
+            raise ValueError(f"The cardinalities of the domain (={d_card}) and codomain (={c_card}) should be equal.")


Suggested change

raise ValueError(f"The cardinalities of the domain (={d_card}) and codomain (={c_card}) should be equal.")

raise ValueError(f"the cardinalities of the domain (={d_card}) and codomain (={c_card}) should be equal")

errors should start with lowercase and not end with punctuations

Done! (Check new commit)

grhkm21 · 2025-03-19T23:20:47Z

src/sage/rings/morphism.pyx

+            raise TypeError(f"{b} fails to convert into the morphism's codomain {C}")
+        F1 = D.base_ring()
+        if F1.prime_subfield() != C.prime_subfield():
+            raise NotImplementedError("The domain's base ring and codomain's prime subfields should match.")


Suggested change

raise NotImplementedError("The domain's base ring and codomain's prime subfields should match.")

raise ValueError("the domain's base ring and codomain's prime subfields should match")

It seems to be this should be a value error?

Done! (Check new commit)

grhkm21 · 2025-03-19T23:58:59Z

src/sage/rings/morphism.pyx

+        from sage.rings.finite_rings.finite_field_base import FiniteField
+        from sage.rings.quotient_ring import QuotientRing_nc
+        if isinstance(self.domain(), QuotientRing_nc) and isinstance(self.codomain(), FiniteField):
+            if self.domain().base_ring().prime_subfield() == self.codomain().prime_subfield():
+                return self._preimage_from_linear_dependence(b)


Can you add a test for this part by the way?

Since _inverse_image_element is a private function, the tests inside should call a public function that eventually calls this private one (for example the rest tests use inverse_image). Is that okay?

Yes. You can add the indirect doctest tag after the test to make the coverage test happy (does anyone care...)

Done! (Check new commit)

grhkm21 · 2025-03-20T00:01:51Z

Also, do you mind elaborating what happens if the map is not bijective but domain and codomain have the same size?

And another question, since ring homs are decided by the action on the generators, would it be more efficient if you cache the result $\beta = f^{-1}(\alpha)$ (if $\alpha$ is a generator of the domain), then $f^{-1}(\alpha^2 + \alpha + 1) = \beta^2 + \beta + 1$ (apply $f$ on both sides to see)

grhkm21 · 2025-03-20T00:29:52Z

src/sage/rings/morphism.pyx

+        from sage.rings.finite_rings.finite_field_base import FiniteField
+        from sage.rings.quotient_ring import QuotientRing_nc
+        if isinstance(self.domain(), QuotientRing_nc) and isinstance(self.codomain(), FiniteField):
+            if self.domain().base_ring().prime_subfield() == self.codomain().prime_subfield():


Suggested change

if self.domain().base_ring().prime_subfield() == self.codomain().prime_subfield():

if self.domain().base_ring().prime_subfield() is self.codomain().prime_subfield():

Sage (especially algebraic) stuff usually inherit from UniqueFactory so is works and is faster (bypasses coercion).

grhkm21 · 2025-03-20T00:40:45Z

One more comment: I realised that this PR is supposed to fix #39690. Can you reflect that in your test? In other words, add a test (or really just a line) showing that f.inverse() works.

rasti37 · 2025-03-20T08:11:24Z

One more comment: I realised that this PR is supposed to fix #39690. Can you reflect that in your test? In other words, add a test (or really just a line) showing that f.inverse() works.

This test should be included in .inverse, ._inverse_image_element or _preimage_from_linear_dependence ?

grhkm21 · 2025-03-20T08:28:25Z

.inverse since it's the public method. IIRC methods starting with an underscore don't need tests officially

github-actions · 2025-03-20T15:01:03Z

Documentation preview for this PR (built with commit 11ae3b1; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

fchapoton and others added 2 commits March 14, 2025 18:10

fixing the ruff linter

e5ca479

added _preimage_from_linear_dependence in morphism.pyx + fixed ruff l…

11ae3b1

…inting based on PR:39699

github-actions bot added the s: needs review label Mar 14, 2025

rasti37 changed the title ~~Compute Preimage for Homomorphism from Quotient Rings to Finite Fields of the same degree~~ Compute preimage for Homomorphism from Quotient Ring to Finite Field of the same degree Mar 14, 2025

rasti37 changed the title ~~Compute preimage for Homomorphism from Quotient Ring to Finite Field of the same degree~~ Implement preimage computation for Homomorphism from Quotient Ring to Finite Field of the same degree Mar 14, 2025

rasti37 changed the title ~~Implement preimage computation for Homomorphism from Quotient Ring to Finite Field of the same degree~~ Implement preimage computation for Homomorphism from Quotient Ring to Finite Field of the same characteristic Mar 18, 2025

grhkm21 suggested changes Mar 19, 2025

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grhkm21 added p: minor / 4 s: needs work c: commutative algebra and removed s: needs review labels Mar 20, 2025

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Implement preimage computation for Homomorphism from Quotient Ring to Finite Field of the same characteristic #39709

Implement preimage computation for Homomorphism from Quotient Ring to Finite Field of the same characteristic #39709

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	ValueError: The cardinalities of the domain (=1024) and codomain (=64) should be equal.
	ValueError: the cardinalities of the domain (=1024) and codomain (=64) should be equal

	raise ValueError(f"The cardinalities of the domain (={d_card}) and codomain (={c_card}) should be equal.")
	raise ValueError(f"the cardinalities of the domain (={d_card}) and codomain (={c_card}) should be equal")

	raise NotImplementedError("The domain's base ring and codomain's prime subfields should match.")
	raise ValueError("the domain's base ring and codomain's prime subfields should match")

	if self.domain().base_ring().prime_subfield() == self.codomain().prime_subfield():
	if self.domain().base_ring().prime_subfield() is self.codomain().prime_subfield():

Implement preimage computation for Homomorphism from Quotient Ring to Finite Field of the same characteristic #39709

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Implement preimage computation for Homomorphism from Quotient Ring to Finite Field of the same characteristic #39709

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rasti37 commented Mar 14, 2025 • edited Loading

Technical Details

How it was fixed?

📝 Checklist

⌛ Dependencies

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