Generalizing the concept of multivariate polynomials and their total degree, we consider the functional degree of maps between (finite) vector spaces and between commutative groups. The functional degree is defined based on a simple idea: it should decrease if a discrete derivative is taken. It may also be seen as the total degree of some kind of integer-valued polynomials. We want to see which special properties functions have if their functional degree is low. In particular, we want to study the distribution of the roots of such functions, which seem to be somehow dispersed over the domain. It may be possible to generalize or strengthen Chevalley and Warning’s Theorem about the set of simultaneous zeros of polynomials in several variables over finite fields. We will also try to generalize the (Quantitative) Combinatorial Nullstellensatz and its corollaries. The language of module theory, finite fields and some simple arithmetic results will be important in this research.
For more information about doctoral scholarship and PhD programme at Xi’an Jiaotong-Liverpool University (XJTLU): Please visit
http://www.xjtlu.edu.cn/en/study-with-us/admissions/entry-requirements
http://www.xjtlu.edu.cn/en/admissions/phd/feesscholarships.html
Requirements:
The candidate should have a first class or upper second class honours degree, or a master’s degree (or equivalent qualification), in Mathematics. Evidence of good spoken and written English is essential. The candidate should have an IELTS score of 6.5 or above, if the first language is not English. This position is open to all qualified candidates irrespective of nationality.
Degree:
The student will be awarded a PhD degree from the University of Liverpool (UK) upon successful completion of the program.
How to Apply:
Interested applicants are advised to email [Email Address Removed] (XJTLU principal supervisor’s email address) the following documents for initial review and assessment (please put the project title in the subject line).
- CV
- Two reference letters with company/university letterhead
- Personal statement outlining your interest in the position
- Proof of English language proficiency (an IELTS score of 6.5 or above)
- Verified school transcripts in both Chinese and English (for international students, only the English version is required)
- Verified certificates of education qualifications in both Chinese and English (for international students, only the English version is required)
- PDF copy of Master Degree dissertation (or an equivalent writing sample) and examiners reports available
Informal enquiries may be addressed to Professor Uwe Schauz ([Email Address Removed]), whose personal profile is linked below,
https://www.xjtlu.edu.cn/en/departments/academic-departments/pure-mathematics/staff/uwe-schauz
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