Richard Baskerville
email: baskerville@acm.org

Jan Pries-Heje

Jan Recker
email: j.recker@qut.edu.au

Principles for Re-Designing Information Systems for Environmental Sustainability

Keywords: Green information systems, environmental sustainability, systems design, theory development, design principles, systems science

Many information systems claim to be "green", meaning in support of environmental sustainability. But at closer look we find that these claims are often unsubstantiated; in other words, many green systems are not making any environment more sustainable. We identify three main root causes. First, the 'environment' is often ill-defined. Second, systems often overlook that 'sustainability' is a targeted function dependent on the goals of some stakeholders, which may include designers, users, organizations, policy makers, society or the planet as a whole. Third, we find that research on green information systems often overlooks conceptualizations such as ecology, environment or sustainability that originate in the sciences of the system, i.e., the basis on which information systems are built. To address these issues we present eight new design principles unique to the development of Green Information Systems that can act as prescriptive coherent design theory for developing information systems that improve environmental sustainability.

Introduction

In responding to increased social, cultural, and legislative pressures that expand the responsibility of firms to increase attention to environmental concerns, chief executives are increasingly turning to information systems (IS), as a solution to assist organizations in transforming to more sustainable entities [START_REF] Gadatsch | Comments on "Green IT: A Matter of Business and Information Systems Engineering[END_REF]. Information systems have been argued to be the greatest force for productivity improvement in the last half century [START_REF] Watson | Information Systems and Environmentally Sustainable Development: Energy Informatics and New Directions for the IS Community[END_REF], and there is great hope that such systems can also help with the global environmental challenge [START_REF] Watson | Information Systems and Environmentally Sustainable Development: Energy Informatics and New Directions for the IS Community[END_REF][START_REF] Elliot | Transdisciplinary Perspectives on Environmental Sustainability: A Resource Base and Framework for IT-Enabled Business Transformation[END_REF].

In response, IS scholars have started to explore the role that information systems provide might play [START_REF] Elliot | Transdisciplinary Perspectives on Environmental Sustainability: A Resource Base and Framework for IT-Enabled Business Transformation[END_REF][START_REF] Melville | Information Systems Innovation for Environmental Sustainability[END_REF][START_REF] Seidel | Sensemaking and Sustainable Practicing: Functional Affordances of Information Systems in Green Transformations[END_REF], and have also taken steps to develop theories and artefacts that show how such systems, often labeled "Green IS", could be built [START_REF] Brewer | WattDepot: An Open Source Software Ecosystem for Enterprise-Scale Energy Data Collection, Storage, Analysis[END_REF][START_REF] Hilpert | Leveraging Green IS in Logistics: Developing an Artifact for Greenhouse Gas Emission Tracking[END_REF][START_REF] Shi | A Decision Support System to Select Suppliers for a Sustainable Supply Chain Based on a Systematic DEA Approach[END_REF].

Our key contention in this paper is that many of the artefacts, theories and conceptions that bear the label "Green IS" do not live up to this name. In other words, we posit that Green IS are not actually and always green, meaning that much of the existing work on Green IS falls short of the proclaimed allure "to assist individuals and organizations to make better, that is, environmentally sustainable decisions, and to enable and effectuate environmentally sustainable work practices rather than environmentally unsustainable ones" [9, p. 2]. In this paper we will discuss three problems with the current conception of Green IS on which this contention is based. We provide a new conceptualization of Green IS and derive a set of six novel design principles that can guide the development of an information system for environmental sustainability. In doing so we take an important step to address the noted Green IS design challenge [START_REF] Watson | Information Systems and Environmentally Sustainable Development: Energy Informatics and New Directions for the IS Community[END_REF][START_REF] Malhotra | Spurring Impactful Research on Information Systems for Environmental Sustainability[END_REF]: How do we build information systems that allow organizational and/or individual actors to perform environmentally sustainable actions and decisions?

2 Related Work

Information Systems and Environmental Sustainability

The scholarly IS discipline has been challenged to provide an understanding how IS can contribute to environmentally responsible human activity [START_REF] Watson | Information Systems and Environmentally Sustainable Development: Energy Informatics and New Directions for the IS Community[END_REF][START_REF] Elliot | Transdisciplinary Perspectives on Environmental Sustainability: A Resource Base and Framework for IT-Enabled Business Transformation[END_REF]. The key assumption is while information technology creates an environmental load due to the electricity required for its operation and the problem of disposing of obsolete hardware, IS can also be used to reduce environmental problems by allowing process and practices to change. This is because, in theory, IS can assist individuals and organizations to make better, environmentally sustainable decisions, and to enable environmentally sustainable work practices.

The studies to date fall in two categories: empirical and design. Empirical studies have, for instance, investigated factors that influence adoption of (any type of) Green IS [e.g., [START_REF] Molla | The Green IT Readiness (G-Readiness) of Organizations: An Exploratory Analysis of a Construct and Instrument[END_REF][START_REF] Chen | An Institutional Perspective on the Adoption of Green IS & IT[END_REF]. Substantive-level studies have conceptualized requirements for some type of Green IS, such as energy systems, or examined specific systems for specific environmental challenges such as energy consumption, greenhouse gas emissions or specific organizational initiatives [e.g., [START_REF] Seidel | Sensemaking and Sustainable Practicing: Functional Affordances of Information Systems in Green Transformations[END_REF][START_REF] Butler | Compliance with Institutional Imperatives on Environmental Sustainability: Building Theory on the Role of Green IS[END_REF].

A second, smaller stream of Green IS research is concerned with the design of information systems for environmental sustainability. This stream has produced a number of instantiations and theories for Green IS design. Reported instantiations include open-source systems for energy data management [START_REF] Brewer | WattDepot: An Open Source Software Ecosystem for Enterprise-Scale Energy Data Collection, Storage, Analysis[END_REF], a greenhouse gas emission tracking system for logistics processes [START_REF] Hilpert | Leveraging Green IS in Logistics: Developing an Artifact for Greenhouse Gas Emission Tracking[END_REF], or an index system for green supplier evaluation [START_REF] Shi | A Decision Support System to Select Suppliers for a Sustainable Supply Chain Based on a Systematic DEA Approach[END_REF]. These efforts have contributed substantive-level design knowledge through the situated implementation of artefacts but they are not presented or developed in a way that the design knowledge becomes more abstract, complete and mature and where they could be termed design theory.

Regarding Green IS design theories, in 2014 two papers were presented at conferences that provide substantive-level design knowledge: they identify specific requirements for a specific type of Green IS design theory, namely an information system for sustainability reporting [START_REF] Hilpert | An Information System Design Theory for Green Information Systems for Sustainability Reporting -Integrating Theory with Evidence from multiple Case Studies[END_REF], and an information system to manage energy consumption [START_REF] Reuter | Identifying the Role of Information Systems in Achieving Energy-Related Environmental Sustainability Using Text Mining[END_REF]. In 2016, Recker [START_REF] Recker | Toward A Design Theory for Green Information Systems[END_REF] proposed a first class-level design theory for green information systems, which postulates that any kind of Green IS must operate on the levels of belief formation, action formation and/or outcome measurement to faithfully belong to the class of systems for environmental sustainability. We will return to this theory below.

Systems and their Environment

For design science, the concept of an environment is not necessarily related to natural environments. It refers quite specifically to the relationship between an artifact and its context. It is important to recognize this relationship in order to avoid too much obsession with the nature of the artifact itself. The context of the artifact, its environment, is the mold in which the artifact must fit sufficiently well to accomplish its goals.

The prospects for developing and applying an artifact to successfully achieve a goal depend on three key elements and their interrelationships. (1) The purpose or goal under which the activity was taken. (2) The characteristics of the artifact itself.

(3) The nature of environment in which the artifact performs.

In Simon's sciences of the artificial view [START_REF] Simon | The Sciences of the Artificial. 3rd ed1996[END_REF], there are two environments. The artifact itself has an inner environment that represents its substance and organization. The second environment is an outer environment constituting the surroundings in which the artifact must operate. Simon refers to the artifact itself as a meeting point or an interface between the inner and the outer environments.

Both the inner and the outer environments can be regarded as systems. For example, a computing artifact will comprise a system of hardware and software. The hardware and software constitute the inner system and the inner environment. If this computing artifact is deployed in an organization, the organization would constitute its outer environment. To the degree that we may consider an organization as a system, this outer environment is also the artifact's outer system. Simon [START_REF] Simon | The Sciences of the Artificial. 3rd ed1996[END_REF] often refers to the "inner system" and the "outer environment".

Because both environments may also be thought of as systems, we must also be concerned with the complexity of the inner environment and the outer environment. A system's complexity is proportional to the amount of information required to describe the system or to resolve any uncertainty associated with the system [START_REF] Klir | Facets of Systems Science[END_REF]. We must be concerned about information at two levels. One level concerns the amount of information flowing across the artifact interface between the two environments. A second level concerns the amount of information necessary to properly specify the artifact's inner environment characteristics in order to match its outer environment across the interface. It also concerns the amount of information necessary to resolve the uncertainty about the artifact's inner environment behavior within the context of its outer environment. Complexity can emerge from the richness of the outer environment. The ability for the artifact to apprehend information about the outer environment across its interface can be seen as a major limiting factor in coping with outer environmental complexity.

Because of the high complexity, it can be impossible to obtain sufficient information to eliminate all uncertainty for many outer environments. Artifacts must often adapt or cope with variations in the outer environment. Such variations could be emergent change or unpredicted environmental conditions. Consequently the inner system may be regarded as a system of organized phenomena capable of attaining the goals in some range of environments. The outer environment is expected to operate across this range. Consequently the outer environment delivers the conditions under which a properly designed inner system will adapt to that outer environment and attain the goals. In this way the conditions of the inner environment are largely determined by the conditions of the outer environment.

The substantive rationality of an inner environment in the way it adjusts to its outer environment constitutes its ability to discover appropriate adaptive inner environment conditions. Thus the limits of the inner environment to discover appropriate adaptive behavior constitute the limits of the inner environments procedural rationality. These are limiting properties of the inner environment appear at the interface and reveal the degree to which the inner environment matches the outer environment. In this way these limiting properties explain artifact defects by tracing them to the inner constraints on adaptivity.

Systems and their Design

Information systems development (ISD) regards the design of processes and products. ISD typically unfolds in a series of stages such as analysis, design, coding and testing. The stages do not have to be carried out sequentially but can be done more or less in parallel. Often each stage operates with a defined notation and will often result in a prescribed artefact, such as a requirements specification or a computer program.

An ISD methodology is a prescribed and prepackaged way of carrying out the development. The package will typically include: (1) activities to be performed; (2) deliverables or artifacts resulting from the activities; and (3) principles for organizing the activities and attaching people to perform the activities. Many ISD methodologies claim to be of generic use. However, an ISD methodology can also be aimed at a specific type of design and development such as Green IS or sustainable systems.

A key question has been whether ISD methodologies were actually used in practice? This question was raised more than 20 years ago when Bødker and Bansler [START_REF] Bødker | A reappraisal of structured analysis: design in an organizational context[END_REF] could not find the prescribed use of structured analysis and design in practice. Following that a growing number of studies suggested that the relationship of methodologies to the practice of information systems development was altogether tenuous [START_REF] Fitzgerald | Systems development methodologies: The problem of tenses[END_REF][START_REF] Wynekoop | Studying system development methodologies: An examination of research methods[END_REF]. At a point it seemed as if the concept of methodology had taken such a dominant role in our thinking about IS design and development that it had become a self-confirming hypothesis; such a thing had to exist. An alternative viewpoint was that IS design and development in reality was emergent and therefor 'amethodical', meaning that there was no predefined sequence, control, or rationality in practice [START_REF] Truex | Amethodical Systems Development: The Deferred Meaning of Systems Development Methods[END_REF].

Three Problems with the Current Conception of Green IS

There are four key assumptions about the design of Green IS that deserve careful distinction.

These assumptions include green, sustainability, ecology, and environment.

The green concept entails the relationship between information systems and the natural environment. It implies a consideration of the relationship between people and nature. It spans issues dealing with the ways in which humankind is deteriorating or destroying the planet Earth. It entails an objective: creating information systems that reduce, or at least do not worsen pollution, biodiversity loss, global warming, the greenhouse effect, and other negative impacts that people create through modern social, economic, and political development [START_REF] Chen | Information Systems and Ecological Sustainability[END_REF].

The sustainability concept usually regards the capacity to meet the needs of the present without compromising the needs of the future [START_REF] Frova | Five Decades of Development Debate on Sustainability[END_REF]. Sustainability can regard the capacity of a design artifact to operate without triggering deterioration of its outer environment. Such a system-based conceptualization of sustainability provides a more general notion that encompasses both green and other forms of sustainability.

The ecology concept usually regards the interrelationships between organisms and their environment. However it can also be applied to systems which have a character that behaves in organic or organic-like ways. In an inner sense, sustainable ecological systems can be regarded as stable. In an outer sense, where the outer environment is itself unstable, sustainable ecological systems can be regarded as resilient [START_REF] Holling | Resilience and Stability of Ecological Systems[END_REF].

The environment concept is also often used in its green sense, meaning the natural environment that is the surroundings or conditions in which a people, animals and plants lives or operate.

On basis of these four assumptions, we identify three problems in the current Green IS literature. One problem is the lack of clarity in distinguishing green as a goal or requirement from green as a characteristic of the artifact. For example, the artifact may be green in an egoist sense of "do no harm" to the environment around it. However, it may at the same time fail to be green in the utilitarian sense of helping to restore our polluted, overheated planet to a previous state that was more desirable: a less polluted, less warm planet.

A second problem is that references to the "environment" are often laden with assumptions, ill-defined and examined in isolation. From a systems perspective, the outer environment of an artifact goes no further than defining the necessary characteristics of its interface, and the functions required of the artifact. However, the environment can and must be seen from many perspectives; natural, economic, organizational, social, ecological, ethical and so on.

A third problem is that systems designers can easily overlook the alternative scope of sustainability requirements. Sustainability is not only relative to inner and outer artifact environments, but it is also a function targeted at the goals of some stakeholders. The sustainability requirements might be defined narrowly according to the goals of certain designers, users, organizations, policy-makers, society or the planet as a whole. Sustainability goals can also conflict. For example, designers, users and organizations may impose sustainability goals that address sustainability of the inner environment, like the organization's stability and resilience. Further, the outer sustainability might restrict the available energy for the artifact, or limit its ability to transfer waste outputs; as a result there could be a loss of stability within the inner system.

A New Design Theory for Green IS

An Illustrative Empirical Case

In September 2015 the Volkswagen (VW) 'Dieselgate' [START_REF]IFIP. IFIP IP3 Denounces Actions of Technologists Who Enabled VW DieselGate Scandal[END_REF] scandal broke loose. The US Environmental Protection Agency went public with the fact that they had found 'defeat' software embedded in diesel engines [START_REF] Holten | The scandal explained[END_REF]. This piece of software was able to detect when a car was being tested, and then it could change the emission of Nitrogen Oxides NO x to the allowed level. In road-tests, however, the emission of NO x was up to 40 times higher [START_REF] Holten | The scandal explained[END_REF].

The green perspective in this case is about NO x . According to GreenFacts [START_REF][END_REF] NO x can decrease lung function and increase the risk of respiratory symptoms such as acute bronchitis, cough and phlegm especially for children. Furthermore, one of the nitrogen oxides namely NO 2 is a trace gas in the atmosphere that absorb the terrestrial radiation leaving the surface of the Earth [START_REF]The Physical Science Basis[END_REF]. Thus high levels of NO x contribute to the 'greenhouse effect'.

When the story broke it was a hot topic for weeks in newspapers all over the world. The International Federation of Information Processing -IFIP -brought a new perspective to the table focusing on the designers or developers. They said in a media release [START_REF]IFIP. IFIP IP3 Denounces Actions of Technologists Who Enabled VW DieselGate Scandal[END_REF]:

"ICT professionals must operate according to a Code of Ethics and should be willing to challenge or even report any order from management that risks the safety of that organization's customers or staff. DieselGate is as much an indictment of the software industry as it is of the VW executives who issued the order for the software to be installed …"

And IFIP went on to state that they saw it as going against professional practice and they pointed to the Milan Declaration [29] for a definition of that practice.

It is easy to see that in this case there are several stakeholders with different perspectives on the question of a sustainable systems design. There is the EPA representing the common opinion that we must reduce the outlet of NO x from cars for health-reasons. There is society-at-large interested in avoiding global warming. There is IFIP that wants to foster a professional profession. There is the carmaker VW that wants to sell cars. And then there are the many users or buyers of the cars. Thus one thing to take into account is the multitude of stakeholders.

The last mentioned stakeholder was the user. The question here is how should they participate?

User participation can be defined as "participation in the system development process by potential users or their representatives" [30, p. 53]. Traditionally user participation has been found to be a major factor in systems' success but there is no clear agreement on the benefits of user participation [START_REF] Ives | User Involvement in MIS Success: A Review of Research[END_REF]. One explanation may be that user participation varies within each stage of the development process [START_REF] Newman | User involvement as an interaction process: A case study[END_REF]. Thus the second issue is who is the user and how can we involve them?

If we now go back to the Dieselgate case we can find an announcement by VW from November 25 where they say they will "… install a small tubular part into some of its engines to help them come into line with European clean air standards" [START_REF] Russel | How Volkswagen Got Away With Diesel Deception[END_REF]. However, in US "…the E.P.A. said it wants to make sure that VW's fix will be effective before ordering a recall. To do so, the agency wants to be able to test diesel cars in its own laboratory and during on-road testing …" [START_REF] Russel | How Volkswagen Got Away With Diesel Deception[END_REF]. These two cites brings in a third dimension; the environment. Europe vs US at one level. Laboratory testing vs. Road testing at another level. Sustaining an economic versus ecological environment at a third level.

As soon as we talk about sustainability or Green IS we need to define; in relation to what environment? We could of course demand that one always took into account the 17 sustainable development goals by the United Nations http://www.un.org/sustainabledevelopment/sustainable-development-goals/ but that would hardly be doable in practice just because of the large number of goals to take into account. Hueting and Reijnders [34, p. 252], for example, discuss goal conflicts and call it "scarcely conceivable for the whole spectrum …". And they continue saying that "Especially, simultaneously realising both … production growth and conservation of the environment, is difficult". Hence, goals may be conflicting, less relevant, and have different importance for different stakeholders. Thus the fourth thing to take into account is sustainability and the relevant environment.

What must Green IS do and how?

Assuming that stakeholders, users, environment and sustainability definitions are given, we can then examine how IS can be designed that can allow organizations and individuals to perform environmentally sustainable work practices and make environmentally sustainable decisions.

Recker [START_REF] Recker | Toward A Design Theory for Green Information Systems[END_REF] argued that IS that are labelled "Green IS" must provide function to support belief formation, action formation and/or outcome assessment as they relate to environmental sustainability. Belief formation captures how psychic states (beliefs, desires, opportunities, etc.) about the natural environment are formed. Action formation describes how psychic states about the natural environment translate to actions that impact the environment. Outcomes describe what the consequences of the actions are. Each of these functions can operate at an individual level, or at an organizational level or both. Recker [START_REF] Recker | Toward A Design Theory for Green Information Systems[END_REF] suggests that an answer to how Green IS can provide functionality lies in examining how information systems provide functional affordances [START_REF] Markus | A Foundation for the Study of IT Effects: A New Look at DeSanctis and Poole's Concepts of Structural Features and Spirit[END_REF], viz., action potentials to users.

Affordances describe the possibilities for goal-oriented action afforded to specified user groups by technical objects such as information systems [START_REF] Markus | A Foundation for the Study of IT Effects: A New Look at DeSanctis and Poole's Concepts of Structural Features and Spirit[END_REF]. They emerge from material properties existent in information systems but emerge at the interaction between user and artifact. Thus, affordances have to be perceived before they can be actualized, and perceiving an affordance does not necessarily mean that users actually realize the offered action possibility [START_REF] Stoffregen | Affordances as Properties of the Animal-Environment System[END_REF].

With these notions in mind, we can delineate how Green IS can be built that afford action potential to belief formation, action formation and/or outcome assessment as they relate to environmental sustainability.

4.3

How should Green IS be designed?

In identifying principles that can guide the development of "true" IS for environmental sustainability on the basis of our arguments above, we took inspiration from the Action Design Research (ADR) methodology [START_REF] Sein | Action Design Research[END_REF] that is prominent in design science research, a research paradigm attempting to develop and evaluate new technology to address practical problems or goals [START_REF] Gregor | Positioning and Presenting Design Science Research for Maximum Impact[END_REF]. We deemed the ADR methodology relevant and applicable because it provides emphasis for development of generalised theory and its focus on the blend of design and action research is fundamental to our emphasis on understanding the empirical domains of stakeholders, users and environment in the development of a Green IS.

Our view of the design principles for Green IS is visualized in Figure 1. Therein, the eight principles are broadly classified into three of the four core stages of the ADR methodology, (1) problem formulation, (2) building, intervention, and evaluation (viz., design), and (3) reflection and learning (viz., theorizing). We omitted the fourth ADR stage, formalization of learning, because this stage centred on generalization and the meta-level design. We explain each principle in turn. Above we have argued that there is a generic requirement that designers make accommodations for the environment in which the designed artefact will operate. In Simon's terms, this is the external interface design of the artefact, such that the environment and the artefact match according to the purpose at hand. For the purpose of our generic approach, it means matching the IS design to the constraints and affordances that are delivered intrinsically by the environment in which the IS stakeholders and users will operate. In terms of design theory, this requirement embodies a specific capability of designers. That is, designers must be able to accommodate the environment.

Principle 2: To identify relevant affordances that must emerge from the relevant stakeholders' interaction with an artefact, it is important to understand their relationship with the ecology, in particular in terms of their goals and abilities.

Principle 3: Delineating the "System" and the "Users" is critical for distinguishing the artefact from its environment. The distinction may be as simple as defining the system as a software driven computing artefact. Such a distinction places the users in the artifact's outer environment, and pushes the social aspects of the ecology across the artifact's interface and out of the inner environment. Alternatively, the users can be regarded as part of the system. Such a distinction places the users within the artifact's inner environment and pulls the social aspects of the ecology across the artifact's interface and into the inner environment.

Principle 4: Once the interface has been delineated, designers can derive the sustainability fit functions for inner environmental sustainability and outer environmental sustainability based on definition of system, environments and user. These sustainability fit functions must incorporate environmental stability factors.

Principle 5: Based on the sustainability fit functions, designers can ensure that the artefact provides proper affordances for the sustainability of both inner & outer environments given the user. Where environments exhibit instability, the affordances must deliver qualities of resilience to both inner and outer environments. In addressing the outer environment, these sustainability affordances will include features that support sustainability of the green ecology.

Principle 6: Sustainability fit functions enable designs that can prioritize the green features. With green sustainability affordances based on the sustainability fit functions, the overall sustainability of the identified (natural) ecology can be given proper priority in the design.

Principle 7: Iteration of the appropriate design science evaluate-and-learn cycle supports refinement and improvement of satisfactory green IS designs. Evaluation covers the judgement or determination of the significance, worth, or quality of the design. Learning covers the acquisition of knowledge or skills throughout the iteration.

Principle 8: Green IS Designs occur in continuing iterations of problem formulation, design, and evaluation/learning that maintain the natural ecological sustainability fit functions as a centrepiece in the design-redesign process. Green IS design is a process of sustainability-guided emergence.

Conclusion

In this paper we contributed a design theory for Green IS that builds on a careful definition of the relevant concepts of stakeholders, system, environment and sustainability. We described our theory in terms of eight novel design principles. Importantly, the class of systems characterized by our design theory has explicit goals of affording environmentally sustainable practices and decisions to users. It allows designer to specify and implement systems that are true to the label "Green IS". With this explicit focus and its requirements, systems in our theory differ from others that are also associated with being "green". Notably, we prescribe that "Green IS" systems must adhere to the requirements set out in this paper.

We note several limitations. First, our research is on the level of theoretical rules and predictions. The development of an expository instantiation and an empirical evaluation of the design principles remains to be completed. Second, much like most other Green IS research we remained focus on the environmental dimension sustainability without regarding interaction effects to other dimensions (e.g., economic or social goals). Still, we believe our research provides a substantial and original contribution to design knowledge in green IS, which has been notably absent to date [START_REF] Watson | Information Systems and Environmentally Sustainable Development: Energy Informatics and New Directions for the IS Community[END_REF][START_REF] Malhotra | Spurring Impactful Research on Information Systems for Environmental Sustainability[END_REF]. Our theory allows both for prescriptiveness and guidance for action as well as for discrimination and identification of boundary conditions. Both is useful in progressing Green IS research because we provide assertions for testing and falsification as well as rules that can delineate the boundaries of the entire field.

Figure 1 :

 1 Figure 1: Design Principles for Green IS Principle 1:Above we have argued that there is a generic requirement that designers make accommodations for the environment in which the designed artefact will operate. In Simon's terms, this is the external interface design of the artefact, such that the environment and the artefact match according to the purpose at hand. For the purpose of our generic approach, it means matching the IS design to the constraints and affordances that are delivered intrinsically by the environment in which the IS stakeholders and users will operate. In terms of design theory, this

Richard Baskerville,

2Jan Pries-Heje & 3Jan Recker


L Busé

M Elkadi
email: elkadi@math.unice.fr

A Galligo
email: galligo@math.unice.fr

Galaad

Intersection and self-intersection of surfaces by means of Bezoutian matrices

Keywords: Algebraic surfaces, intersection locus, self-intersection locus, resultants, Bezoutian

The computation of intersection and self-intersection loci of parameterized surfaces is an important task in Computer Aided Geometric Design. We address these problems via four resultants with separated variables; two of them are specializations of general multivariate resultants and the two others are specializations of determinantal resultants. We give a rigorous study in these four cases and provide new formulas in terms of Bezoutian matrix.

Introduction

A common representation of surfaces in Solid Modeling and Computer Aided Geometric Design (CAGD) uses parameterized patches, i.e. images of maps of the form

Φ : [0, 1] × [0, 1] → R 3 (u, v) → Φ(u, v) = Φ 1 (u, v) Φ 0 (u, v) , Φ 2 (u, v) Φ 0 (u, v) , Φ 3 (u, v) Φ 0 (u, v) (1)
where Φ 0 , Φ 1 , Φ 2 , Φ 3 are polynomials with real coefficients and bi-degree (m, n). When Φ 0 (u, v) is a non-zero constant (independent of u and v), Φ is called a polynomial map, otherwise Φ is called a rational map. These patches are encountered in many applications [START_REF] Farin | Curves and surfaces for computer aided geometric design[END_REF], and spline surfaces are made by gluing them together.

There are many articles presenting methods and algorithms to intersect two patches [START_REF] Sederberg | Loop detection in surface patch intersections[END_REF][START_REF] Krishnan | An efficient surface intersection algorithm based on lower-dimensional formulation[END_REF][START_REF] Hosaka | Modeling of Curves and Surfaces in CAD/CAM[END_REF][START_REF] Patrikalakis | Surface-to-surface intersections[END_REF] and to compute self-intersections loci [START_REF] Galligo | A sampling algorithm for parametric surface selfintersection[END_REF][START_REF] Lasser | Self-intersections of parametric surfaces[END_REF][START_REF] Maekawa | Computation of self-intersections of offsets of bezier surface patches[END_REF][START_REF] Volino | Collision and self-collision detection: Efficient and robust solutions for highly deformable surfaces[END_REF][START_REF] Thomassen | Self-intersection problems and approximate implicitization[END_REF]. These operations still need to be improved and computer algebra techniques can help. Among them, generalized resultant (for instance sparse resultant) techniques have been successfully employed [START_REF] Khetan | The resultant of an unmixed bivariate system[END_REF][START_REF] Khetan | Sylvester A-resultants for bivariate polynomials with planar Newton polygons (extended abstract)[END_REF][START_REF] Busé | Using projection operators in computer aided geometric design[END_REF].

The computation of self-intersection of a patch or intersection of two patches are important problems in CAGD; they were the main topics of the European project GAIA II [START_REF]GAIA II. European project[END_REF]. Several strategies have been developed to address these problems: either via multivariate resultants [START_REF] Busé | Using projection operators in computer aided geometric design[END_REF], or via special case study [START_REF] Galligo | The geometry of bicubic surfaces and splines[END_REF], or via approximate implicitization [START_REF] Dokken | Approximate implicitization[END_REF][START_REF] Thomassen | Self-intersection problems and approximate implicitization[END_REF], or via numerical methods [START_REF] Galligo | A sampling algorithm for parametric surface selfintersection[END_REF].

Numerical methods algorithms are very efficient, however they often rely on samplings, and have the weakness common to these techniques: if the size of a loop in the (self-)intersection locus is smaller than the step-size of the sampling, it might become invisible and the program is unable to compute it.

There are heuristics to choose the step-size, but they do not really overcome this difficulty.

Therefore, symbolic (or semi-numeric) methods are essential in order to solve completely these tasks. For that purpose, the parameterizations should be also algebraic. The most commonly used algebraic representations in CAGD are the so-called NURBS and particularly the polynomial patches.

In this paper, we will present new tools developed for this purpose. Namely specific sparse resultants and corresponding Bezoutian formulas allowing to compute, in each case presented below, a bivariate polynomial which is the equation of a plane projection of the double points locus. This polynomial is represented as the determinant of a small matrix, and thus it is useful for further post-processing.

For this setting, we prepare the equations for the elimination procedure in order to get a special format that we call with separated variables. Then we show that the obtained systems have nice properties that we further study and exploit.

We construct an unmixed bivariate resultant Res(P 1 , P 2 , P 3 ) where the three polynomials P i (x, y) = f i (x) -g i (y), and apply it to the computation of implicit equations of self-intersection and intersection loci of polynomial surfaces which is an important task in CAGD and Solid Modeling. In practice, this resultant is computed via a Bezoutian matrix, and it is related to (but different from) the toric resultant studied in [START_REF] Khetan | Sylvester A-resultants for bivariate polynomials with planar Newton polygons (extended abstract)[END_REF][START_REF] Khetan | The resultant of an unmixed bivariate system[END_REF]. In the case of rational surfaces, we will use an anisotropic resultant to address the intersection problem and a determinantal resultant in the presence of base points to address the selfintersection problem. For each case, we show how to build a square matrix (namely a Bezoutian matrix) whose determinant is the expected resultant.

We implemented our algorithms in Maple; an illustration and comparisons with the Maple package bires written by Amit Khetan are presented. We will show that the size of the matrix used in our approach to compute the different resultants are smaller than those given by Khetan's method [START_REF] Khetan | The resultant of an unmixed bivariate system[END_REF].

The methodology that we will use along this paper to obtain an elimination formula adapted to our special situations can be summarized as follows. Starting from a given polynomial system E, the first step is to "homogenize" it (i.e. to embed it into an irreducible projective variety) with the condition that the obtained homogeneous system E h do not have base-points. Then, by the general theory of resultants, this system can be seen as a particular instance of a general resultant which is not identically zero. Moreover, the particular instance is always geometrically irreducible because, due to the hypothesis that E h is base-point free, the incidence variety is a vector bundle. However, it need not to be algebraically reduced, but this property can be shown through well chosen specializations. This approach will be applied to the four different cases corresponding respectively to intersection and self-intersection of polynomial and rational parameterizations.

The paper is organized as follows. The next section provides a short overview on the Bezoutian matrix and its main property. In section 3, we precise the applications in CAGD and set the equations and the elimination problems for intersection and self-intersection. In section 4, we go back to our main CAGD applications and make explicit how our resultants can be used. We sketch also the algorithms and present examples. Sections 5 and 6 are more theoretical and they contain the proofs of our results. In section 5, we prove that in our situation, the specialization process is valid and that resultants (which are anisotropic) can be computed by nice Bezoutian formulas. In section 6, we describe an adapted version of the determinantal resultant and prove that these resultants can also be computed by simple Bezoutian formulas.

Overview on the Bezoutian matrix

In this short section we recall a well known matrix construction to eliminate variables from polynomial systems. We will apply it to prove formulas for bivariate resultants that will be used to solve the intersection and the selfintersection problems that we address.

) := f 0 (x) ∂ 1 f 0 (x, y) • • • ∂ n f 0 (x, y) . . . . . . . . . . . . f n (x) ∂ 1 f n (x, y) • • • ∂ n f n (x, y) (2)
where

∂ i f j (x, y) := f j (y 1 , . . . , y i-1 , x i , . . . , x n ) -f j (y 1 , . . . , y i , x i+1 , . . . , x n ) x i -y i . Set Bez f 0 ,...,fn (x; y) = α,β c α,β x α y β with c α,β ∈ R.
Fixing an order on the monomials, the matrix (c α,β ) α,β whose rows are indexed by the y β such that there exists α with c α,β = 0, and columns are indexed by the x α such that there exists β with c α,β = 0, is called the Bezoutian matrix of f 0 , . . . , f n .

Notice that we can substitute x by y in the first column of the determinant Bez f 0 ,...,f n (x; y) since

f j (x) -f j (y) = n i=1 (x i -y i )∂ i f j (x, y).
Observe also that the Bezoutian matrix does not have a null row or a null column.

The Bezoutian matrix is an interesting tool in elimination theory because it allows to eliminate several variables at once in polynomial systems. More precisely, suppose given a system f 0 (c, x) = • • • = f n (c, x) = 0 where c denotes a collection of parameters and x denotes a collection of variables. Then, under suitable conditions of genericity [START_REF] Busé | Generalized resultants over unirational algebraic varieties[END_REF]Theorem 2.2] it can be shown that there exists an irreducible polynomial in the parameters c which vanishes for a value c 0 of c if and only if the zero locus of f 0 (c 0 , x)

= • • • = f n (c 0 , x) = 0 is strictly larger than the zero locus of f 0 (c, x) = • • • = f n (c, x) = 0.
This irreducible polynomial is called the generalized resultant of f 0 (c, x), . . . , f n (c, x) with respect to the variables x. Moreover it divides the determinant of any maximal minor of the Bezoutian matrix of f 0 (c, x), . . . , f n (c, x) [START_REF] Busé | Generalized resultants over unirational algebraic varieties[END_REF]Theorem 3.4] (see also [START_REF] Busé | Using projection operators in computer aided geometric design[END_REF]Theorem 3.7] for general statements). Therefore, if ∆(c) stands for the determinant of a maximal minor of the Bezoutian matrix of f 0 (c, x), . . . , f n (c, x), then the vanishing of ∆ at a given value c 0 of c is a necessary condition for the vanishing of the generalized resultant of f 0 (c, x), . . . , f n (c, x) at c 0 . In general, this condition is not sufficient; for instance, in the univariate case the determinant of the Bezoutian matrix of two polynomials with different degrees differs from their resultant by a power of the leading coefficient of one of these polynomials. We refer the reader to [5, §4] for more examples and illustrations.

In this paper, we will consider four special classes of bivariate resultants for which we will prove that the determinants of the associated Bezoutian matrices are exactly equal to the corresponding generalized resultants.

The main results

In this section we state four theorems addressing two important tasks in CAGD: the intersection and the self-intersection problems for polynomial and rational surfaces. The proofs of these results are rather technical as they require some knowledge in algebraic geometry and rely on the use of specific resultants; they will be given in sections 5 and 6. Note also that the condition sufficiently generic required in the following results will also be completely described at that time.

Intersection problem

First we describe a particular projection of the pre-image curve (i.e the curve in the parameter space which maps to the intersection locus). We consider the intersection of two patches S 1 and S 2 given by parameterizations Φ(u, v) and Ψ(s, t) with the same features as (1) that we view in the projective space P 3 . The set S 1 ∩ S 2 corresponds to the quintuple of parameters (u, v, s, t, λ), with λ = 0, such that Φ(u, v) = λΨ(s, t). This set form a curve C in R 5 that we can describe (i.e. give the topology and witness points on each component) via a well chosen projection on a plane

C 1 = {(u, t) : ∃ (v, s, λ) with λ = 0 and Φ(u, v) = λΨ(s, t)}.
Our geometric task is equivalent to describe the curve C 1 . Its implicit equation is obtained by eliminating v, s, λ in the system Φ(u, v) = λΨ(s, t) of 4 equations defining C.

In the polynomial case (i.e. Φ 0 (u, v) = Ψ 0 (s, t) = 1), the question reduces to the elimination of 2 variables v and s in a system of 3 polynomial equations in 4 variables (there is no λ).

Theorem 2 (see subsection 5.1) Suppose given two sufficiently generic parameterized polynomial surfaces of bi-degree (m, n):

(u, v) → Φ 1 (u, v), Φ 2 (u, v), Φ 3 (u, v) , (s, t) → Ψ 1 (s, t), Ψ 2 (s, t), Ψ 3 (s, t) .
The implicit equation of the projection of their intersection locus onto the plane (u, t) equals the determinant of the mn × mn Bezoutian matrix, given by the formula (2), of Φ 1 (u, v) -Ψ 1 (s, t), Φ 2 (u, v) -Ψ 2 (s, t), Φ 3 (u, v) -Ψ 3 (s, t) viewed as polynomials in v and s.

In the rational case, we have the following result: Theorem 3 (see subsection 6.1) Suppose given two sufficiently generic parameterized rational surfaces of bi-degree (m, n):

(u, v) → Φ 1 (u, v) Φ 0 (u, v) , Φ 2 (u, v) Φ 0 (u, v) , Φ 3 (u, v) Φ 0 (u, v) , (s, t) → Ψ 1 (s, t) Ψ 0 (s, t) , Ψ 2 (s, t) Ψ 0 (s, t) , Ψ 3 (s, t) Ψ 0 (s, t) .
The implicit equation of the projection of their intersection locus onto the plane (u, t) equals the determinant of the mn × mn matrix B(u, t) defined by

U B(u, t)V t = 1 (v -v 1 )(s -s 1 ) Φ 0 (u, v) Φ 0 (u, v 1 ) Ψ 0 (s, t) Ψ 0 (s 1 , t) Φ 1 (u, v) Φ 1 (u, v 1 ) Ψ 1 (s, t) Ψ 1 (s 1 , t) Φ 2 (u, v) Φ 2 (u, v 1 ) Ψ 2 (s, t) Ψ 2 (s 1 , t) Φ 3 (u, v) Φ 3 (u, v 1 ) Ψ 3 (s, t) Ψ 3 (s 1 , t)
where U and V denote the monomial vectors

U := [s i v j , i = 0 . . . m -1, j = 0 . . . n -1], V := [s 1 i v 1 j , i = 0 . . . m -1, j = 0 . . . n -1].
Observe that U B(u, t)V t in Theorem 3 is a polynomial in v, v 1 , s, s 1 , u, t.

Self-intersection problem

We start with a parameterization Φ given by (1) that we view in P 3 and, as in the previous subsection, we will describe a well chosen projection of the pre-image curve of the self-intersection locus.

The set of double points of Φ can be characterized by a curve

C = {(u 1 , v 1 , u 2 , v 2 , λ) : (u 1 , v 1 ) = (u 2 , v 2 ), λ = 0, Φ(u 1 , v 1 ) = λΦ(u 2 , v 2 )} in R 5
. It can also be described in a simpler way by a curve of its well chosen projection (generically 1 to 1) C 1 on R 2 :

C 1 = {(u, v); ∃ (k, l, λ) : (k, l) = 0, λ = 0, and Φ(u, v + k) = λΦ(u + l, v)}.
Our geometric task is equivalent to describe the curve C 1 . Its implicit equation is obtained by the elimination of l, k, λ in the system of 4 equations defining C 1 .

If the variables u, v are fixed, we remark that the point (k, l, λ) = (0, 0, 1) is a base-point (i.e. a solution which is independent of u and v) of the system

Φ(u, v + k) -λΦ(u + l, v) = 0 defining C 1 .
We will exploit this observation to develop an adapted resultant for the self-intersection problem in subsections 5.2 and 6.2.

In the polynomial case, we have the following result:

Theorem 4 (see subsection 5.2) Suppose given a sufficiently generic parameterized polynomial surface of bi-degree (m, n):

(u, v) → Φ 1 (u, v), Φ 2 (u, v), Φ 3 (u, v) .
The implicit equation of the projection of its self-intersection locus onto the plane (u, v) equals the determinant of the (mn -1) × (mn -1) Bezoutian matrix B(u, v) defined by the formula (2):

U B(u, v)V t = Bez Φ 1 (u,v+k)-Φ 1 (u+l,v),Φ 2 (u,v+k)-Φ 2 (u+l,v),Φ 3 (u,v+k)-Φ 3 (u+l,v) (k, l; k 1 , l 1 ) with U := l i k j , (i, j) ∈ {0, . . . , m -1} × {0, . . . , n -1} \ {0, 0}
, and

V := l 1 i k 1 j , (i, j) ∈ {0, . . . , m -1} × {0, . . . , n -1} \ {0, 0} .
In the rational case, we have:

Theorem 5 (see subsection 6.2) Suppose given a sufficiently generic parameterized rational surface of bi-degree (m, n):

(u, v) → Φ 1 (u, v) Φ 0 (u, v) , Φ 2 (u, v) Φ 0 (u, v) , Φ 3 (u, v) Φ 0 (u, v) .
The implicit equation of the projection of its self-intersection locus onto the plane (u, v) equals the determinant of the (mn -1) × (mn -1) Bezoutian matrix B(u, v) defined by the formula (2):

U B(u, v)V t = 1 (k -k 1 )(l -l 1 ) Φ 0 (u, v + k) Φ 0 (u, v + k 1 ) Φ 0 (u + l, v) Φ 0 (u + l 1 , v) Φ 1 (u, v + k) Φ 1 (u, v + k 1 ) Φ 1 (u + l, v) Φ 1 (u + l 1 , v) Φ 2 (u, v + k) Φ 2 (u, v + k 1 ) Φ 2 (u + l, v) Φ 2 (u + l 1 , v) Φ 3 (u, v + k) Φ 3 (u, v + k 1 ) Φ 3 (u + l, v) Φ 3 (u + l 1 , v)
where U and V denote the monomial vectors

U := l i k j , (i, j) ∈ {0, . . . , m -1} × {0, . . . , n -1} \ {0, 0} , V := l 1 i k 1 j , (i, j) ∈ {0, . . . , m -1} × {0, . . . , n -1} \ {0, 0} . Notice that U B(u, v)V t in Theorem 5 is a polynomial in k, k 1 , l, l 1 , u, v.

Comments and illustrations

The four theorems given in Section 3 are parts of a general approach to the intersection and self-intersection problems of algebraic surfaces that we now illustrate. For this purpose, we focus on the polynomial self-intersection problem; the three other cases can be treated similarly.

Consider a surface given by

(u, v) ∈ R 2 → Φ 1 (u, v), Φ 2 (u, v), Φ 3 (u, v) ∈ R 3
a polynomial parameterization of bi-degree (m, n) and define the polynomials

P i (u, v, l, k) := Φ i (u, v + k) -Φ i (u + l, v) , i = 1, 2, 3, of bi-degree (m, n) in the variables (l, k) with polynomial coefficients in (u, v) of bi-degree (m, n). Set R(u, v) := Res P 1 (u, v, l, k), P 2 (u, v, l, k), P 3 (u, v, l, k)
where the right hand side in this equality is a resultant operator that will be precised in subsection 5.2. In order to draw the portion of the projection of the self-intersection curve of the parameterized surface, say D, contained in [0, 1] 2 , we should find a point on each of its connected components. A point on a component crossing the border of the square is easily found, so the difficulty is essentially concentrated on the internal loops. A common way to find a point on such a loop is to compute all the critical points of D on an axis, e.g. the u-axis. This amounts to solve the bivariate system

R(u, v) = ∂R ∂u (u, v) = 0.
Once we get a point on each connected component of D contained in [0, 1] 2 , the computer algebra part of the process is completed. Then a numerical marching algorithm can be used to compute the points (u, v + k, u + l, v) ∈ [0, 1] 4 needed to draw efficiently the pre-image curve or any of its plane projection. Note that a part of this process can be done by exploiting the kernel of the matrix which is used to compute the resultant R(u, v): for instance, if the couple (u, v) satisfies det B(u, v) = 0, the vector U in Theorem 4 is in the kernel of the transpose of this matrix, so we can often obtain k and l as the quotient of two sub-minors of B(u, v) (see [START_REF] Busé | Inversion of parameterized hypersurfaces by means of subresultants[END_REF] for precise results).

An illustrative example. Let S be a parameterized surface of bidegree (2, 2) given by:

Φ 1 (s, t) := 2 s 2 -t 2 s 2 -5 t s 2 + 7 s + 3 t 2 s + s t -1 + 6 t 2 -2 t, Φ 2 (s, t) := 9 t 2 s 2 -5 t s 2 + 2 t 2 s + 4 s t + s -7 t 2 + 9 t, Φ 3 (s, t) := t s 2 + 2 s t -2 t 2 .
To find the self-intersection locus of S, we express that this parameterization takes the same value at the two pairs of parameters (s, t) = (u, v + k) and (s, t) = (u + l, v). By Theorem 4 we form the Bezoutian of the three polynomials Φ i (u, v+k)-Φ i (u+l, v) for i = 1, 2, 3 (see Defintion 1) which can be written as U B(u, v)V t , with U = (l, k, lk) and V = (l 1 , k 1 , l 1 k 1 ). The matrix B(u, v) is a 3 × 3 symmetric matrix of rank 3. Its determinant D(u, v) is of bi-degree [START_REF] Farin | Curves and surfaces for computer aided geometric design[END_REF][START_REF] Farin | Curves and surfaces for computer aided geometric design[END_REF] in (u, v) and of total degree 14. At a generic point (u, v) on the curve

D(u, v) = 0, B(u, v) is of rank 2, therefore the kernel of t B(u, v) = B(u, v
) is generated by a vector of type U . By Cramer's rules, the entries of U are given by 2 × 2-minors of B(u, v), and after simplifications we find that l and k are given by two rational functions L(u, v) and K(u, v) of total degree 6 in (u, v).

The graph of the curve D(u, v) = 0 is drawn in Figure 1. We notice that at the point (u, v) = (-1.05, -0.049) three local components of this curve intersect (see Figure 2). This can be checked by a formal algebraic computation. Now the graphs spanned by u, v+K(u, v) and by u+L(u, v), v , when (u, v) ranges on the graph of D(u, v), have also three local components intersecting at the two images of (u, v) = (-1.05, -0.049). Geometrically, on the surface S this corresponds to a singularity of the self-intersection curve similar to that of the Roman Steiner's surface depicted in [START_REF] Coffman | The algebra and geometry of Steiner and other quadratically parametrizable surfaces[END_REF]. Comparison with Khetan's formulas. We end this section by comparing the four new elimination formulas we gave in Section 3 with the formulas given by Khetan to compute bivariate sparse resultants [START_REF] Khetan | The resultant of an unmixed bivariate system[END_REF]. As far as we know, it is the lonely known matrix based method that produces exact elimination formulas. We implemented the constructions of matrices presented in Section 3 in Maple and used the Maple package bires written by Khetan. Khetan's formulas are designed for bivariate resultants of any three polynomials, whereas our formulas are only available for bivariate resultants of polynomials with separated variables. However, since we took into account this particular structure, our formulas compared most of the time favorably to the ones given by Khetan. Moreover, we observed sometimes that Khetan's hybrid matrix is not full rank in some cases whereas the determinant of the matrix we propose gives exactly the projection of the intersection and self-intersection curve that we are looking for.

The following table presents some experiments. In the first column, we indicate the bi-degree of a randomly chosen rational surface. The second column gives the size of the matrix constructed by bires to compute the self-intersection locus of this rational surface, whereas the last column yields the size of the matrix obtained by Theorem 5 to solve the same problem. The size of matrices obtained by Theorem 5 are significantly smaller than the ones given by Khetan's approach. It follows that the computation of the selfintersection locus is highly increase because the more time-consuming step is by large the computation of determinants of these matrices (their construction is negligible compared to the computation of their determinants). Finally, we mention that Khetan's formulas may fail to be non-zero; for instance, for randomly chosen rational surface of bi-degree (2,2), his approach returns a 11 × 11 matrix which is not full rank.

From now on, we focus on the proofs of the four theorems given in Section 3.

Bivariate resultants with separated variables

In this section, we suppose given three univariate polynomials of degree m ≥ 1

f i (x) := a i,0 x m + a i,1 x m-1 + • • • + a i,m-1 x + a i,m ∈ Z[a i,0 , . . . , a i,m ][x]
and three univariate polynomials of degree n ≥ 1

g i (y) := b i,0 y n + b i,1 y n-1 + • • • + b i,n-1 y + b i,n ∈ Z[b i,0 , . . . , b i,n ][y].
Hereafter we will consider resultants of the three following bivariate polynomials with separated variables

P i (x, y) := f i (x) -g i (y) ∈ A[x, y], i = 1, 2, 3,
where A := Z[(a i,j ) i=1,2,3;j=0,...,m , (b i,j ) i=1,2,3;j=0,...,n ] denotes the universal coefficients ring of these polynomials.

The dense case

This case will be applied to solve the intersection problem for polynomial surfaces.

To compute the classical resultant (sometimes called Macaulay's resultant) of the polynomials P 1 (x, y), P 2 (x, y) and P 3 (x, y), we first need to homogenize them to the projective space by introducing a new variable. But it is clear that this resultant is identically zero, since this homogenization produces at least one base-point (i.e. a common root) in P 2 . To overcome this problem, a natural idea is to change variables and compute the classical resultant of the polynomials P 1 (x n , y m ), P 2 (x n , y m ) and P 3 (x n , y m ) which is not identically zero. However, this resultant could be non reduced: it could be a power of an irreducible polynomial. As described in [START_REF] Jouanolou | Le formalisme du résultant[END_REF], its reduced part corresponds to a resultant which is obtained by a (special) quasi-homogenization of P 1 (x, y), P 2 (x, y) and P 3 (x, y) to a weighted projective space a P 2 . We will rely on this construction which has been fully studied in [START_REF] Jouanolou | Le formalisme du résultant[END_REF][START_REF] Jouanolou | Résultant anisotrope, compléments et applications[END_REF].

Recall that the weighted projective plane a P 2 with a weight a = (a 0 , a 1 , a 2 ) ∈ N 3 over a field K is just the quotient space by the following relation: two points (x 0 , x 1 , x 2 ) and (y 0 , y 1 , y 2 ) in K 3 \ {0} are equivalent if there exists λ ∈ C * such that (x 0 , x 1 , x 2 ) = (λ a 0 y 0 , λ a 1 y 1 , λ a 2 y 2 ). For instance, the weighted projective plane with the weight (1, 1, 1) is simply the usual projective plane. For more details on this topic we refer the reader to [START_REF] Beltrametti | Introduction to the theory of weighted projective spaces[END_REF].

In general, the specialization of a given resultant operator to a set of nongeneric polynomials may gives either an identically zero polynomial or a non irreducible polynomials on the restricted parameter space. However we will show that the specializations of some resultants are correct for our classes of non-generic polynomials but this requires a precise analysis and a rigorous proof, given below.

In our setting, introducing a new variable z, we homogenize P 1 (x, y), P 2 (x, y) and P 3 (x, y) to the weighted projective space a P 2 with homogeneous coordinates (x, y, z) of weight (n, m, 1). We get, for i = 1, 2, 3, the quasi-homogeneous polynomials in A[x, y, z]:

f h i (x, z) := m j=0 a i,j x m-j z nj , g h i (y, z) := n j=0
b i,j y n-j z mj and P h i (x, y, z) := f h i (x, z) -g h i (y, z) of weight mn. We denote by Resa P 2 the anisotropic resultant of three polynomials in a P 2 . Hereafter we consider the graded polynomial ring A with the usual grading: each of its variables has weight 1.

Proposition 6

The specialization Resa P 2 (P h 1 , P h 2 , P h 3 ) of the anisotropic resultant is a non-zero polynomial. It is homogeneous of degree mn with respect to the coefficients of each polynomial P 1 , P 2 and P 3 , and hence has total degree 3mn. Moreover it is an irreducible element in A.

Proof. By specializing P h

1 to x m , P h 2 to y n and P h 3 to z mn the anisotropic resul-tant R := Resa P 2 (P h 1 , P h 2 , P h 3 ) equals 1 and it is hence non-zero. Moreover, by the general result of [START_REF] Jouanolou | Le formalisme du résultant[END_REF][START_REF] Jouanolou | Résultant anisotrope, compléments et applications[END_REF], R is homogeneous with respect to the coefficients of each polynomial P 1 , P 2 and P 3 of degree mn.

The difficult point is to prove the irreducibility of R. We first prove that R is geometrically irreducible (i.e. it is a power of an irreducible polynomial). To do this we consider the incidence variety W := {((x : y : z), (a i,j , b i,j )) :

P h 1 = P h 2 = P h 3 = 0} ⊂ a P 2 × A 3(m+n+2)
and its two canonical projections π 1 and π 2 onto the first and the second factor respectively. By construction π 1 gives W a structure of vector bundle over a P 2 which is an irreducible variety. We deduce that W is itself an irreducible variety and therefore that its image by π 2 is irreducible in the affine space A 3(m+n+2) . This image has the same support than the zero locus of R and hence it follows that R is a certain power, say e, of an irreducible polynomial. Finally, to see that e = 1 we need to prove that the projection of W by π 2 is bi-rational onto its image. This done by considering the specialization φ defined, for generic α i s and β j s by

φ(P h 1 ) = m i=1 (x -α i z n ) , φ(P h 2 ) = n j=1 (y -β j z m ) , φ(P h 3 ) = x m -y n
which sends the anisotropic resultant R to

φ(R) = Resa P 2 ( m i=1 (x -α i z n ), n j=1 (y -β j z m ), x m -y n ) = m,n i,j=1 Resa P 2 (x -α i z n , y -β j z m , x m -y n ) = m,n i,j=1 (α n i -β m j ).
(these two last equalities rely on properties of anisotropic resultants proved in [START_REF] Jouanolou | Le formalisme du résultant[END_REF]). Since the above product contains the irreducible and reduced factor (α 1 -β 1 ), e must be equal to 1. 2

We just proved the existence of a non-zero particular resultant which is adapted to our setting. The next task is to show that this resultant can be computed as the determinant of a certain square matrix, namely a Bezoutian matrix that we now describe.

It is easy to check that the Bezoutian (see Definition 1) associated to the polynomials P 1 (x, y), P 2 (x, y) and P 3 (x, y):

Bez P 1 ,P 2 ,P 3 (x, y; x 1 , y 1 ) = - f 1 (x) -g 1 (y) f 1 (x)-f 1 (x 1 ) x-x 1 g 1 (y)-g 1 (y 1 ) y-y 1 f 2 (x) -g 2 (y) f 2 (x)-f 2 (x 1 )
x-x 1 g 2 (y)-g 2 (y 1 ) y-y 1

f 3 (x) -g 3 (y) f 3 (x)-f 3 (x 1 )
x-x 1 g 3 (y)-g 3 (y 1 ) y-y 1 and to deduce, after manipulations on the columns, that this Bezoutian has a particular symmetry property (which is not true in general), namely

Bez P 1 ,P 2 ,P 3 (x 1 , y 1 ; x, y) = Bez P 1 ,P 2 ,P 3 (x, y; x 1 , y 1 ).

The (x 1 , y 1 )-monomial support of this Bezoutian (i.e. the set of monomials x 1 i y 1 j having non-zero coefficients) is {0, . . . , m -1} × {0, . . . , n -1}, and have hence exactly mn elements. By the symmetry property, the (x, y)-monomial support is the same. Therefore the Bezoutian matrix B of P 1 , P 2 , P 3 is, in this case, a square matrix of size mn and we have:

Theorem 7
The determinant of the Bezoutian matrix B of P 1 , P 2 , P 3 equals the anisotropic resultant Resa P 2 (P h 1 , P h 2 , P h 3 ) up to a sign1 in A.

Proof. From the discussion in Section 2, Resa P 2 (P h 1 , P h 2 , P h 3 ) divides det(B). Moreover, it is easy to see by Definition 1 that each entry of B is a polynomial in A which has total degree 3 and is homogeneous of degree 1 with respect to the coefficients of each polynomial P 1 , P 2 and P 3 . Therefore, det(B) ∈ A is homogeneous with respect to the coefficients of each P 1 , P 2 and P 3 of degree mn. Now, if det(B) is non-zero, it equals the anisotropic resultant Resa P 2 (P h 1 , P h 2 , P h 3 ) up to a non-zero constant. Considering the specialization φ defined by φ(P 1 ) = x m , φ(P 2 ) = y n and φ(P 3 ) = 1, it is easy to check that

φ Bez P 1 ,P 2 ,P 3 (x, y; x 1 , y 1 ) = x m -x m 1 x -x 1 × y n -y n 1 y -y 1 ,
and that φ(det(B)) = ±1. So det(B) = 0 and the theorem is proved. 2

Note that, in more general setting, other formulations of the anisotropic resultant as a determinant of a square matrix has already been established in [START_REF] Jouanolou | Résultant anisotrope, compléments et applications[END_REF], and also in [START_REF] Khetan | The resultant of an unmixed bivariate system[END_REF] as a sparse resultant. Theorem 7 is a simple expression in the particular case of polynomial systems with separated variables.

The residual case

This case is adapted to the determination of self-intersection loci of polynomial surfaces.

We keep the notations of the subsection 5.1, but now we assume that the polynomials f i (x) and g i (x), for all i = 1, 2, 3, have the same constant term (which is the case in the self-intersection problem), or in other words that P 1 (0, 0) = P 2 (0, 0) = P 3 (0, 0) = 0.

Thus we have for i = 1, 2, 3, a i,m = b i,n and we view P 1 , P 2 and P 3 as polynomials in A [x, y], where A := Z[(a i,j ) i=1,2,3;j=0,...,m , (b i,j ) i=1,2,3;j=0,...,n-1 ] is the universal coefficients ring.

Similarly to the previous subsection, we introduce a new variable z and homogenize the polynomials to the weighted projective space a P 2 with coordinates (x, y, z) of weight (n, m, 1). Then we have, for i = 1, 2, 3,

f h i (x, z) := m j=0 a i,j x m-j z nj , g h i (y, z) := a i,m z mn + n-1 j=0 b i,j y n-j z mj and P h i (x, y, z) := f h i (x, z) -g h i (
x, z) are quasi-homogeneous of weight mn. However, in this new setting, the anisotropic resultant Resa P 2 (P h 1 , P h 2 , P h 3 ) is degenerate because P h 1 , P h 2 and P h 3 vanish at the same point P := (0 : 0 : 1) independently of their coefficients.

In order to deal with this base-point and get a non-identically zero resultant, we consider the blow-up π P : a P2 → a P 2 of the weighted projective space along the point P whose definition ideal is I P := (x, y), with exceptional divisor D. Following techniques developed in [START_REF] Busé | Etude du résultant sur une variété algébrique[END_REF][START_REF] Busé | Resultant over the residual of a complete intersection[END_REF] (see also [START_REF] Busé | Using projection operators in computer aided geometric design[END_REF] for a quick overview), we obtain the so-called residual resultant, and denoted by Res a P2 (P h 1 , P h 2 , P h 3 ). It provides a necessary and sufficient condition for the zero locus in a P 2 of the system P h 1 (x, y, z) = P h 2 (x, y, z) = P h 3 (x, y, z) = 0 to be, scheme-theoretically, strictly bigger than the point P . For more details, we refer the reader to [START_REF] Busé | Etude du résultant sur une variété algébrique[END_REF][START_REF] Busé | Using projection operators in computer aided geometric design[END_REF][START_REF] Busé | Resultant over the residual of a complete intersection[END_REF].

Proposition 8

The residual anisotropic resultant Res a P2 (P h 1 , P h 2 , P h 3 ) is a nonzero polynomial. It is homogeneous of degree mn -1 with respect to the coefficients of each polynomial P 1 , P 2 , P 3 and of total degree 3mn -3. Moreover it is an irreducible element in A .

Proof. The proof is similar to that of Proposition 6, as it amounts to consider an anisotropic residual resultant instead of a simple anisotropic one.

The geometric irreducibility of R := Res a P2 (P h 1 , P h 2 , P h 3 ) follows from two facts. First, the blow-up a P2 is an irreducible projective variety, since a P 2 is itself a projective irreducible variety. Second, the incidence variety

W = {( t, (a i,j , b i,j )) : P h 1 ( t) = P h 2 ( t) = P h 3 ( t) = 0} ⊂ a P2 × A 3(m+n+1)
is a vector bundle over a P2 . Here P h i denotes the virtual transform of P h i by the blow-up π P (i.e. π * P (P h i ) divided only one time by the exceptional divisor D).

As P is a smooth point in a P 2 , the self-intersection number a P2 D 2 of D equals -1. Since each P h i is quasi-homogeneous of degree mn, the multi-degree formula for residual resultants (see [START_REF] Busé | Etude du résultant sur une variété algébrique[END_REF]) shows that R is homogeneous, with respect to the coefficients of each P i , of degree mn + a P2 D 2 = mn -1.

Finally, to prove that R = 0 and it is not a power of an irreducible polynomial, we exhibit a specialization of P 1 , P 2 and P 3 such that the corresponding specialization of R is non-zero and contains an irreducible and reduced factor, as we did in the proof of Proposition 6. For that purpose, we compute

Res a P2 (x m-1 i=1 (x -α i z n ), y n-1 i=1 (y -β i z m ), x m -y n ) = m-1,n-1 i,j=1 (α n i -β m j ). 2
Having defined the anisotropic residual resultant, we now describe how to compute it as the determinant of a Bezoutian matrix.

We proceed exactly as in the dense case, by considering the Bezoutian matrix of the polynomials P 1 , P 2 , P 3 . The only difference is that a i,m = b i,n for i = 1, 2, 3, which implies that the (x, y)-monomial support (resp. the (x 1 , y 1 )monomial support) of this Bezoutian does not contain the monomial 1 = x 0 y 0 (resp. 1 = x 0 1 y 0 1 ) and hence contains only mn -1 elements. Therefore, we get a square matrix B of size mn -1 and we have:

Theorem 9
The determinant of the Bezoutian matrix B of P 1 , P 2 , P 3 is exactly the residual anisotropic resultant Res a P2 (P h 1 , P h 2 , P h 3 ) up to a sign in A .

Proof. From the discussion in Section 2, Res a P2 (P h 1 , P h 2 , P h 3 ) divides det( B). It is straightforward to check that each entry of B is a polynomial in A which has total degree 3 and is homogeneous of degree 1 with respect to the coefficients of each P 1 , P 2 and P 3 . Therefore det( B) ∈ A is homogeneous with respect to the coefficients of each polynomial P 1 , P 2 and P 3 of degree mn -1. Now, considering the specialization φ defined by φ(P 1 ) = x m + x, φ(P 2 ) = y n and φ(P 3 ) = x -y, we obtain that φ Bez P 1 ,P 2 ,P 3 (x, y; x 1 , y 1 ) is equal to

x m -x m 1 x -x 1 + 1 y n + (x -y) y n -y n 1 y -y 1 -(x m + x) y n -y n 1 y -y 1 .
This implies that φ(det( B)) = ±1 and concludes the proof. 2

6 Bivariate determinantal resultants with separated variables

With the strategy described in Section 3, the computation of the intersection locus of two rational surfaces requires the elimination of variables x, y, λ in a polynomial system of the form

φ i (x) = λψ i (y), i = 0 . . . 3 and λ = 0,
where the φ i 's (resp. the ψ i 's) are univariate polynomials in the variable x (resp. y) of given degree m ≥ 1 (resp. n ≥ 1). This is equivalent to the fact that the 4 × 2 matrix defined by the φ i 's and ψ i 's has rank 1 at some point (x, y). This leads to the notion of determinantal resultant introduced and studied in [START_REF] Busé | Etude du résultant sur une variété algébrique[END_REF]. Therefore, to study the intersection and self-intersection problems for rational surfaces we need an extension of tools used in the polynomial case to the determinantal setting.

Introducing new variables x and ȳ, we can put this system into a (bi)-projective context to first eliminate the variable λ. Then, we eliminate the two couples of homogeneous variables (x, x) and (y, ȳ) from the bi-homogeneous polynomial system

φ h 0 (x, x) : φ h 1 (x, x) : φ h 2 (x, x) : φ h 3 (x, x) = ψ h 0 (y, ȳ) : ψ h 1 (y, ȳ) : ψ h 2 (y, ȳ) : ψ h 3 (y, ȳ) ∈ P 3 ,
where, for i = 0, 1, 2, 3, deg(

φ h i ) = m ≥ 1 and deg(ψ h i ) = n ≥ 1.
This elimination problem does not correspond to the projection of a complete intersection incidence variety, as in Section 5, then the previous theory of resultants does not apply. However, there is a generalization of the previous approach where the incidence variety is a determinantal one, and one can define a determinantal resultant (see [START_REF] Busé | Resultants of determinantal varieties[END_REF]). It turns out that this can be applied in our setting: this elimination problem fits into the class of determinantal polynomial systems (which is much larger than the class of complete intersection ones).

We re-formulate the elimination problem as the necessary and sufficient condition on the parameters of the polynomials φ h i and ψ h i , i = 0 . . . 3, so that there exists a point (x : x), (y : ȳ) ∈ P 1 × P 1 which satisfies rank

   φ 0 (x, x) φ 1 (x, x) φ 2 (x, x) φ 3 (x, x) ψ 0 (y, ȳ) ψ 1 (y, ȳ) ψ 2 (y, ȳ) ψ 3 (y, ȳ)    < 2.
Let us fix the notations. From now on, we suppose given four univariate polynomials of degree m ≥ 1, i = 0 . . . 3,

φ i (x) := a i,0 x m + a i,1 x m-1 + • • • + a i,m-1 x + a i,m ∈ Z[a i,0 , . . . , a i,m ][x]
and their corresponding homogenizations φ h i (x, x) ∈ Z[a i,0 , . . . , a i,m ][x, x], as well as four univariate polynomials of degree n ≥ 1, i = 0 . . . 3,

ψ i (y) := b i,0 y n + b i,1 y n-1 + • • • + b i,n-1 y + b i,n ∈ Z[b i,0 , . . . , b i,n ][y]
and their corresponding homogenizations

ψ h i (y, ȳ) ∈ Z[b i,0 , . . . , b i,n ][y, ȳ]
. We denote by A := Z[(a i,j ) i=0,...,3;j=0,...,m , (b i,j ) i=0,...,3;j=0,...,n ] the universal coefficients ring of these polynomials.

We will use the theory of determinantal resultants developed in [START_REF] Busé | Resultants of determinantal varieties[END_REF]. In particular, we will provide new formulas to compute determinantal resultants with separated variables using Bezoutian matrices.

The dense case

Let us denote by X := P 1 × P 1 , O X the vector bundle of rational functions on X, and consider two vector bundles E := O 4 X and F := O X (m, 0) ⊕ O X (0, n). Taking the coefficients of polynomials φ i 's and ψ i 's in C, this system corresponds to a map of vector bundles on X, ρ : E → F , whose matrix is given by

M (x, x), (y, ȳ) :=    φ 0 (x, x) φ 1 (x, x) φ 2 (x, x) φ 3 (x, x) ψ 0 (y, ȳ) ψ 1 (y, ȳ) ψ 2 (y, ȳ) ψ 3 (y, ȳ)    .
Following [START_REF] Busé | Resultants of determinantal varieties[END_REF], the determinantal resultant associated to this morphism exists and corresponds to the necessary and sufficient condition on φ i 's and ψ i 's coefficients so that there exists a point (x 0 : x0 ), (y 0 : ȳ0 ) ∈ P 1 ×P 1 satisfying rank M (x 0 : x0 ), (y : ȳ0 ) ≤ 1.

We denote this determinantal resultant, which is an element in A defined up to a sign, by Res X (M). In this case, we have the following proposition: Proposition 10 The determinantal resultant Res X (M) is a non-zero irreducible polynomial in A. Moreover, for each j ∈ {0, . . . , 3}, it is a homogeneous polynomial with respect to the coefficients of φ j and ψ j of degree mn, and it is hence a homogeneous element in A of degree 4mn.

Proof. First, we will justify the existence of Res X (M). Set H = Hom(E, F ). Although the vector bundle H is not very ample on X, the proof of Theorem 1 in [START_REF] Busé | Resultants of determinantal varieties[END_REF] applies. In this proof, the very ampleness hypothesis is used to show that the projection from the incidence variety W to the projectivized parameter space P(H):

W = {(x, φ) ∈ X × P(H) : rank(φ(x)) ≤ 1} → P(H)
is bi-rational onto its image (which is called the resultant variety). The argument is the following: given a zero-dimensional sub-scheme z of P 1 × P 1 of degree two (that is two distinct points or a double point), the locus of matrices φ in P(H) of rank lower or equal to 1 on z is of co-dimension twice the co-dimension of matrices of rank lower or equal to 1 on only one smooth point (which is here 3). In our particular situation, this property remains true, even if H is not very ample.

To obtain the claimed partial degrees we use the intersection theory as developed in [START_REF] Fulton | Intersection theory[END_REF]: given an integer j ∈ {0, . . . , 3} we know from [START_REF] Busé | Resultants of determinantal varieties[END_REF] that the degree of Res X (M) with respect to the coefficients of φ j and ψ j equals the coefficient of the monomial α j in the coefficient of t 3 in the expansion of the series

- 4 i=1 (1 -α i t) c t (F ) = - 1 c t (F ) (1 -(α 1 + α 2 + α 3 + α 4 )t + • • • )
where c t (F ) denotes the Chern polynomial of F . We deduce that this degree is exactly the degree of the coefficient of t 2 in the series 1/c t (F ). Denoting by H 1 (resp.H 2 ) the class of the generic point in the first (resp. second) factor P 1 of X, we have

c t (F ) = c t O X (m; 0) c t O X (0; n) = (1 + mH 1 )(1 + nH 2 ) = 1 + mH 1 + nH 2 + mnH 1 H 2 .
It follows that

1 c t (F ) = 1 -c 1 (F )t + c 1 (F ) 2 -c 2 (F ) t 2 + • • •
from which we obtain that the degree of Res X (M) with respect to the coefficients of φ j and ψ j equals

X c 1 (F ) 2 -c 2 (F ) = X (mH 1 + nH 2 ) 2 -mnH 1 H 2 = 2mnH 1 H 2 -mnH 1 H 2 = X mnH 1 H 2 = mn.
2 Remark 11 We recall from [START_REF] Busé | Resultants of determinantal varieties[END_REF] that Res X (M) is, for all (p, q) ∈ N 2 such that p ≥ 3m -1 and q ≥ 3n -1, the determinant of the following graded part of the Eagon-Northcott complex

S[-3, -1] (p-3m;q-n) ⊕ S[-2, -2] (p-2m;q-2n) ⊕ S[-1, -3] (p-m;p-3n) ∂ 2 -→ S[-2, -1] 4 (p-2m;q-n) ⊕ S[-1, -2] 4 (p-m;q-2n) ∂ 1 -→ S[-1, -1] 6 (p-m;q-n) ∧ 2 ρ --→ S[0, 0] (p;q)
where S is the ring

S := (Z[(a i,j ) i=0,...,3,j=0,...,m ] ⊗ Z Z[(b i,j ) i=0,...,3,j=0,...,n ]) ⊗ Z (Z[x, x] ⊗ Z Z[y, ȳ])
which is equipped (via tensor products) with two (bi-graded) gradings that we denote by S[-, -] (-,-) . In particular, Res X (M) vanishes if and only if the rank of the 9mn × 24mn matrix

S[-1, -1] 6 (2m-1;2n-1) ∧ 2 ρ --→ S[0, 0] (3m-1;3n-1)
drops. Moreover, from this result we can deduce that Res X (M) ∈ A is homogeneous with respect to the coefficients of (φ i ) i=0,...,3 (resp.(ψ i ) i=0,...,3 ) of degree 2mn.

We now turn to the computation of this determinantal resultant in our setting. We consider the four polynomials

P i (x, y, λ) := φ i (x) -λψ i (y), i = 0 . . . 3,
and we compute the Bezoutian Bez P 0 ,...,P 3 (x, y, λ; x 1 , y 1 , λ 1 ) which equals -λ φ 0 (x) φ 0 (x)-φ 0 (x 1 )

x-x 1 ψ 0 (y) ψ 0 (y)-ψ 0 (y 1 )

y-y 1 φ 1 (x) φ 1 (x)-φ 1 (x 1 ) x-x 1 ψ 1 (y) ψ 1 (y)-ψ 1 (y 1 ) y-y 1 φ 2 (x) φ 2 (x)-φ 2 (x 1 ) x-x 1 ψ 2 (y) ψ 2 (y)-ψ 2 (y 1 ) y-y 1 φ 3 (x) φ 3 (x)-φ 3 (x 1 )
x-x 1 ψ 3 (y) ψ 3 (y)-ψ 3 (y 1 ) y-y 1 . This Bezoutian does not depend on the variable λ 1 , and we can eliminate λ from it by defining Bez P 0 ,...,P 3 (x, y; x 1 , y 1 ) := Bez P 0 ,...,P 3 (x, y, λ; x 1 , y 1 , λ 1 ) λ .

It is clear that Bez P 0 ,...,P 3 (x, y; x 1 , y 1 ) = Bez P 0 ,...,P 3 (x 1 , y 1 : x, y). The (x 1 , y 1 )monomial support (and hence also its (x, y)-monomial support) of this Bezoutian is {0 . . . m -1} × {0 . . . n -1}. Therefore we can construct a square matrix B of size mn from it and we have:

Theorem 12
The determinant of the matrix B equals to the determinantal resultant Res X (M) in A up to a sign.

Proof. From the discussion in Section 2, Res X (M) divides det(B ). Since they have the same degree as homogeneous elements in A, we deduce that the claimed result will be proved if we show that det(B ) equals ±1 for a given specialization. For this, we consider the specialization ρ defined by

ρ(φ 0 ) = x m , ρ(φ 1 ) = 0, ρ(φ 2 ) = 0, ρ(φ 3 ) = 1, ρ(ψ 0 ) = 0, ρ(ψ 1 ) = y n , ρ(ψ 2 ) = 1, ρ(ψ 3 ) = 0.
It is straightforward to check that ρ(det(B )) = ±1 (the sign depends on the ordering chosen on monomials to construct B ). 2

The residual case

With the same notations as in subsection 6.1, we now assume that for i = 0 . . . 3, the polynomials φ i (x) and ψ i (y) have the same constant term (which is the case for the self-intersection problem); that is

φ i (0) = a i,m = b i,n = ψ i (0) for i = 0, 1 , 2, 3.
The ring A := Z[(a i,j ) i=0,1,2,3;j=0,...,m , (b i,j ) i=0,1,2,3;j=0,...,n-1 ] denotes the universal coefficients ring.

We remark that the determinantal resultant Res X (M) ∈ A in Proposition 10 is identically zero in this case because the matrix M is always of rank less or equal to 1 at P = (0 : 1), (0 : 1) ∈ P 1 × P 1 . So, we are no longer considering all the morphisms E → F , but only those having rank 1 at this point P .

Similarly to what we did in subsection 5.2, we can overtake this difficulty by considering the blow-up π P : X → X = P 1 × P 1 of X along P . Then the morphism π * P (E) → π * P (F ) have generically rank 2 outside the exceptional divisor D and rank 1 on D. Therefore, for such a general morphism, the cokernel Q fits in the exact sequence

π * P (E) |D → π * P (F ) |D → Q → 0
and is a vector bundle of rank 1 over D. It follows that the kernel F of the canonical surjective morphism π * P (F ) → Q → 0 is also a vector bundle2 over X and has the same rank than F . We have the exact sequence

0 → F → π * P (F ) → Q → 0. ( 3
)
It turns out that the morphism π * P (E) → π * P (F ) factors through F and the residual determinantal resultant we are looking for is the determinantal resultant of vector bundles Ẽ := π * P (E) and F over X. We will denote it by Res X (M). It gives a necessary and sufficient condition so that the determinantal locus in X corresponding to the condition rank    φ 0 (x, x) φ 1 (x, x) φ 2 (x, x) φ 3 (x, x) ψ 0 (y, ȳ) ψ 1 (y, ȳ) ψ 2 (y, ȳ) ψ 3 (y, ȳ)    < 2 is, scheme-theoretically, strictly bigger than the point P .

Proposition 13

The residual determinantal resultant Res X (M) is a non-zero irreducible polynomial in A . Moreover, for each integer j ∈ {0, . . . , 3}, it is homogeneous with respect to the coefficients of φ j and ψ j of degree mn -1, and hence it is a homogeneous element in A of degree 4mn -4.

Proof. The proof of the irreducibility of Res X (M) is exactly the same than the one given in Proposition 10 using the general theory developed in [START_REF] Busé | Resultants of determinantal varieties[END_REF]. Given an integer j ∈ {0, . . . , 3}, the degree of Res X (M) with respect to the coefficients of φ j and ψ j equals the degree of the coefficient of t 2 in the series 1/c t ( F ). From the exact sequence (3) we deduce that c t ( F )c t (Q) = c t (F ). Moreover, using the definition of M and Q we have Q O D , so we deduce that

1 c t ( F ) = c t (O D ) c t (F ) = 1 c t (F )c t (O X (-D)) = 1 c t (F ) × 1 1 -Dt = 1 -c 1 (F )t + (c 1 (F ) 2 -c 2 (F ))t 2 + • • • 1 + Dt + D 2 t 2 + • • • .
Hence the degree of Res X (M) with respect to the coefficients of φ j and ψ j equals

X D 2 -c 1 (F )D + c 1 (F ) 2 -c 2 (F ) = X c 1 (F ) 2 -c 2 (F ) + X D 2 = mn -1.
As a consequence, the residual determinantal resultant Res X (M) is an homogeneous element in A of degree 4(mn -1) = 4mn -4.

2

To compute this residual determinantal resultant we use, as before, Bezoutian matrices. With the same notations and computations as in subsection 6.1, we build the Bezoutian matrix B of size mn × mn. However in this more particular setting, the couple (0, 0) (associated to the monomial 1) is not in the (x, y)-monomial support and not also in the (x 1 , y 1 )-monomial support. In this way, we consider the mn -1 × mn -1 sub-matrix B of B corresponding to erase the row and the column which defines the monomial 1.

Theorem 14

The determinant of the matrix B equals the residual determinantal resultant Res X (M) in A up to a sign.

Proof. From the discussion in Section 2, Res X (M) divides det( B ). Since these two polynomials have the same degree as homogeneous elements in A , we deduce that the claimed result is proved if we show that det( B ) equals ±1 for a given specialization. To do this, we consider the specialization ρ defined by ρ(φ 0 ) = x m + x, ρ(φ 1 ) = 0, ρ(φ 2 ) = x, ρ(φ 3 ) = 1, ρ(ψ 0 ) = 0, ρ(ψ 1 ) = y n , ρ(ψ 2 ) = -y, ρ(ψ 3 ) = 1, and it is straightforward to check that ρ(det( B )) = ±1 (the sign depends on the ordering chosen on monomials to construct the matrix B ). 2

Conclusion

In this paper we presented a re-formulation of computer algebra problems related to the determination of the self-intersection and intersection loci of polynomial surfaces patch and in the more general case of rational patches used for NURBS. We gave simple and efficient formulas to compute the implicit equation of certain projections of these loci. The next step is to study a numeric adaptation of the original approach described in this paper.

Fig. 1 .Fig. 2 .

 12 Fig. 1. A view of the double-point locus

  Bi-degree Size of bires matrix Size of Bezoutian matrix (

It is possible to fix the sign in this equality by requiring an ordering on the (x, y)-monomial and the (x 1 , y 1 )-monomial supports of B.

this is a consequence of the fact that Q has rank 1. The construction of F is known as an elementary transformation of the vector bundle F (see[START_REF] Maruyama | Elementary transformations in the theory of algebraic vector bundles[END_REF] for more details).

Acknowledgments

This work was partially supported by the French Research Agency (ANR Gecko) and the European Network of Excellence Aim@Shape (IST NoE 506766). The authors thanks André Hirschowitz for helpful discussions on elementary transformations of vector bundles.