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AutocompleteAbout the same103 / 280 · beu-f38fecc670bf731a

Numerical Elimination: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets

Mathematics · 2601.04383v3

FLOWING EVIDENCE BENCHMARK

How do we compare writing with and without cards?

At the same writing position, the same model produces one first result with cards and one without. All 280 pairs enter the overall anonymized review, which assesses whether a decision not to continue fits the draft.

Overall: 138 with-card wins, 26 ties, 107 no-card wins, 5 pairs where neither was usable, and 4 inconclusive judgments. Both sides wrote text in 218 pairs; 48 had a deliberate stop and 14 included an invalid output.

SAME DRAFT · WITH OR WITHOUT CARDS

01 · FIXED WRITING POSITION

MANUSCRIPT

Same writing position ▌

Matched autocomplete at the same draft position

02 · TWO MATCHED VERSIONS

SHARED BY BOTH

Draft, writing task, and generation model

A · With cards

Relevant literature cards supplied

B · Without cards

No literature cards supplied

Generation model

Same model and version

A → first output

B → first output

Anonymized AI judge

First outputs are anonymized as X and Y

Continuation review: accuracy, fit to the writing task and usability
Returns: X preferred / tie / Y preferred / both unusable

Review again with the answer order swapped

HOW DOES BLIND JUDGING WORK?

① Anonymize both outputs
The AI judge sees the same draft and both first outputs without knowing which received cards.

② Compare and swap order
The judge applies task-specific criteria in X/Y and then Y/X order.

③ Review disagreements
A third pass resolves disagreements. Incomplete reviews remain in the denominator.

How are positions and cards selected?

We selected 40 writing positions in each of seven fields, for 280 comparisons. The position and relevant literature cards were set before each comparison, so both versions began from the same draft.

The main result retains every frozen position, including places where cards might not add a useful next move. This avoids selecting cases based on their outputs, but 49% describes the whole cohort, not a win rate where evidence clearly had an opportunity to help. This cohort did not record that opportunity label for each position before generation, so those two groups cannot yet be separated reliably.

CARD USE · 140/280

Reported card use in continuations with text

The model reports whether a card affected its continuation. In 140 of 280 tests it produced continuation text and reported card use. The previous count of 150 included 10 cases without continuation text and has been corrected.

COMPARISON · 70 / 7 / 30

Which written version was better?

Of the 107 reported-use cases where both versions produced text, the blinded AI judge preferred the with-card version 70 times, found 7 about the same, and preferred the without-card version 30 times.

How are the two table percentages calculated?

ADOPTION COVERAGE

140reported use with text
÷
280all positions
=
50%

WITH-CARD VERSION PREFERRED

70with cards better
÷
107comparable tests
=
65%

The percentages use different groups: 50% counts continuations with text reporting card use among all 280 tests; 65% covers the 107 such cases with text on both sides. The generating model reports card use, while an anonymized AI judge compares writing quality.

Card use is self-reported by the generator. The anonymized AI judge sees the draft and source snippets, so source-aware preference remains possible; there was no independent human review. The overall result describes these 280 frozen positions, not a win probability for any writing task.

Manuscript writing position

Text excerpt · not a PDF page

Research manuscript · excerpt

Numerical Elimination: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets

Introduction / related work

Hpx,tqis nonsingular alongtPp0,1s. Explicitly, Hpx,tq“p1´tq ˜ x2 2`x 1x2`1 2x2`x 1 ¸ `γt ˜ x2 1´1 x2´1 ¸ . The start pointsp1,1qandp´1,1qare nonsingular solutions ofHpx,1q“0. Both corresponding paths are trackable and converge to the solutionsp2,´1qandp´2,1qofHpx,0q“0. This shows that the intersection of the zero set ofFwith the planeVpb´1qistp2,1,´1q,p´2,1,1qu.♢ When only a subset of the solutions to a system of polynomials or rational functions is known, monodromy provides a method for finding additional solutions [7, 18, 22, 26, 47]. In this approach, we view our target system as a member of a parametric family of systems, and let the parameters vary continuously along a loop in parameter space, while numerically tracking a known solution. When the loop returns to its starting point, the endpoint of the tracked path is again a solution of the original system, but not necessarily the same one. This induces an action of the fundamental group of the complement of the branch locus on the fiber of solutions over a base parameter, called themonodromy action. The image of this action inside the permutation group of the solutions is themonodromy group. See [25] for further discussion of

↓ Next paragraph was generated here

The highlighted text is near the writing position. This is a layout of the evaluation input, not a PDF screenshot.

A matched comparison at the same writing positionOverall AI review: About the same

Both arms use the same model, task and manuscript context. Only the retrieved passages differ. The first recorded output from each arm is shown below.

Read overall AI review rationale

This rationale combines reviews with different display orders.

Order-swapped judges disagreed. Pass 1: Both outputs complete the sentence fragment 'See [25] for further discussion of' with appropriate completions. Output A ('the monodromy group and its computation') is more precise and better fits the manuscript's argument. The draft has just defined 'monodromy group' as the key object of interest, and the preceding paragraph focuses on the monodromy action/group as a specific concept. Output A's definite article 'the' correctly refers back to this specific monodromy group just defined, maintaining terminological consistency. Output B's plural 'monodromy groups' and broader phrase 'in the context of numerical algebraic geometry' introduces unnecessary generality and slight scope drift—the draft is discussing a specific construction, not surveying multiple monodromy groups or the broader field. Output A is tighter, more accurate to the local argument, and requires no repair upon insertion. Pass 2: Both outputs appropriately complete with 'completion' actions, recognizing that the sentence 'See [25] for further discussion of' is an incomplete citation that needs only a brief object. Output A ('monodromy groups and their computation in the context of numerical algebraic geometry') is more precise and useful: it specifies 'monodromy groups' (plural, matching the preceding discussion of the monodromy action and monodromy group as concepts), adds 'their computation' (directly relevant to the paper's focus on numerical methods), and crucially includes 'in the context of numerical algebraic geometry' which fits the manuscript's evident focus on numerical/algorithmic approaches (evidenced by prior discussion of tracking paths, nonsingular solutions, and the citations to numerical methods). Output B ('the monodromy group and its computation') is grammatically acceptable but less informative—using singular 'the monodromy group' is slightly awkward after the text just defined both 'monodromy action' and 'monodromy group' as distinct concepts, and omitting 'numerical algebraic geometry' misses an opportunity to reinforce the paper's methodological context. The extra scope in A is appropriate, not excessive, and better preserves the argument's direction.

Evidence

Same model and position · retrieved passages supplied · first output

monodromy groups and their computation in the context of numerical algebraic geometry.

Passages supplied to the Evidence version

Card 1 · Numerical computation of Galois groups

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric…

Read full passage excerpt

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric group. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators while the other gives information on its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.

Card 2 · Solving the area-length systems in discrete gravity using homotopy continuation

Area variables are intrinsic to connection formulations of general relativity, in contrast to the fundamental length variables prevalent in metric formulations. Within 4D discrete gravity, particularly based on triangulations, the area-length system establishes a relationship between area variables…

Read full passage excerpt

Area variables are intrinsic to connection formulations of general relativity, in contrast to the fundamental length variables prevalent in metric formulations. Within 4D discrete gravity, particularly based on triangulations, the area-length system establishes a relationship between area variables associated with triangles and the edge length variables. This system is comprised of polynomial equations derived from Heron's formula, which relates the area of a triangle to its edge lengths. Using tools from numerical algebraic geometry, we study the area-length systems. In particular, we show that given the ten triangular areas of a single 4-simplex, there could be up to 64 compatible sets of edge lengths. Moreover, we show that these 64 solutions do not, in general, admit formulae in terms of the areas by analyzing the Galois group, or monodromy group, of the problem. We show that by introducing additional symmetry constraints, it is possible to obtain such formulae for the edge lengths …

Card 3 · Visualizing Planar and Space Implicit Real Algebraic Curves with Singularities

We present a new method for visualizing implicit real algebraic curves inside a bounding box in the $2$-D or $3$-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing space curve in higher-dimensional space. Since the topology of…

Read full passage excerpt

We present a new method for visualizing implicit real algebraic curves inside a bounding box in the $2$-D or $3$-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing space curve in higher-dimensional space. Since the topology of a curve near a singular point of it is not numerically stable, we trace only the curve outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is $ε$-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the projection of the curve in $\mathbb{R}^2$, which is important for applications such as solving bi-parametric polynomial systems. The algorithm starts by computing three types of key points of the curve, namely the intersection of the curve with small spheres centered at singular points, regular critical points of every connected components of th …

Card 4 · Galois/monodromy groups for decomposing minimal problems in 3D reconstruction

We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from…

Read full passage excerpt

We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases--3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras--where the decomposition and symmetries may be naturally understood in terms of the Galois/monodromy group. We then show how our framework can be applied to novel problems from absolute and relative pose estimation. For instance, we discover new symmetries for absolute pose problems involving mixtures of point and line features. We also describe a problem of estimating a pair of calibrated homographies between three …

Card 5 · On computing local monodromy and the numerical local irreducible decomposition

Similarly to the global case, the local structure of a holomorphic subvariety at a given point is described by its local irreducible decomposition. Following the paradigm of numerical algebraic geometry, an algebraic subvariety at a point is represented by a numerical local irreducible decomposition…

Read full passage excerpt

Similarly to the global case, the local structure of a holomorphic subvariety at a given point is described by its local irreducible decomposition. Following the paradigm of numerical algebraic geometry, an algebraic subvariety at a point is represented by a numerical local irreducible decomposition comprised of a local witness set for each local irreducible component. The key requirement for obtaining a numerical local irreducible decomposition is to compute the local monodromy action of a generic linear projection at the given point, which is always well-defined on any small enough neighborhood. We characterize some of the behavior of local monodromy action of linear projection maps under analytic continuation, allowing computations to be performed beyond a local neighborhood. With this characterization, we present an algorithm to compute the local monodromy action and corresponding numerical local irreducible decomposition for algebraic varieties. The results are illustrated using s …

Model-reported card use

The model did not report that any card materially affected this completion.

No Evidence

Same model and position · no retrieved passages · first output

the monodromy group and its computation.

Earlier card-use comparison

The earlier diagnostic compared only cases with reported card use and text on both sides; its scope differs from the overall AI review above.

This case does not have both with-card continuation text and a model-reported use record. It remains among all 280 tests but does not enter the reported-use comparison.

The passages are input material; their presence does not mean an output used them or that they support every claim. Draft and source passages are excerpted; no simulated scores or PDF appear here.