Analysis: Extra Components in Resultants for Elimination

In the field of algebraic geometry, resultants are often used for the elimination of variables in systems of polynomial equations. However, a common issue arises when the variety being considered contains components of dimension larger than the expected dimension. In such cases, the resultant vanishes, making it unreliable for the desired elimination.

In an attempt to address this issue, J. Canny proposed a solution involving symbolically perturbing the system before computing the resultant. This perturbed resultant introduces additional artefact components that are loosely related to the geometry of the variety of interest. While this solves the problem of vanishing resultants, it poses a new challenge of removing these extra components from the final result.

J.M. Rojas offered a solution to this challenge by suggesting that taking the greatest common divisor of the results obtained from two different perturbations can effectively remove the unwanted components. By considering multiple perturbations, it becomes possible to discern the persistent extra components and eliminate them from the final elimination result.

However, in this paper, the authors delve deeper into this construction and investigate the nature of these extra components that persist even after taking different perturbations. The analysis reveals that these persistent extra components can only come from either singularities or positive-dimensional fibers within the variety.

Implications and Future Directions

This finding has significant implications for future research in the field of elimination theory. By identifying the sources of persistent extra components, researchers can better understand the underlying geometric properties of varieties and develop more sophisticated techniques for their elimination.

One potential direction for future research is investigating the relationship between singularities and extra components in resultants for elimination. Understanding how these singularities contribute to the presence of persistent extra components can provide valuable insights into the structure of varieties and guide the development of more efficient elimination algorithms.

Additionally, the connection between positive-dimensional fibers and persistent extra components opens up new avenues for exploration. Investigating the properties of these fibers and their impact on the elimination process can lead to the development of novel techniques for removing unwanted components from resultants.

In conclusion, this paper sheds light on an important issue in elimination theory and presents a promising approach for addressing it. The identification of persistent extra components as originating from singularities or positive-dimensional fibers paves the way for further advancements in the field, allowing researchers to refine elimination techniques and gain deeper insights into the geometry of algebraic varieties.

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