Skip to main content
MathOverflow

Return to Question

edited tags
Source Link
Dan
Dan
  • 111
  • 4

Cross-posted from Math.SE.

I've been wondering recently about results for irreducibility that use the "additive structure" of the polynomial ring at hand. For instance, can we say anything about the irreducibility of a sum of two irreducible polynomials that satisfy some conditions? This seems like an additive number theory problem but in a different context. I searched the literature for stuff along these lines but did not find much. Am I missing something obvious (like maybe I can be asking the same questions in the underlying ring) or is this just a question with sparse literature around it? Any papers involving similar questions would be extremely appreciated!

Here's an example: $(x^{2}+2x+1)p(x)+x$ seems to be irreducible for "almost all" p(x) with nonzero constant term. Unsure how I'd show this. Here's a paper that kind of has the flavor of problems that I'm interested in.

I've been wondering recently about results for irreducibility that use the "additive structure" of the polynomial ring at hand. For instance, can we say anything about the irreducibility of a sum of two irreducible polynomials that satisfy some conditions? This seems like an additive number theory problem but in a different context. I searched the literature for stuff along these lines but did not find much. Am I missing something obvious (like maybe I can be asking the same questions in the underlying ring) or is this just a question with sparse literature around it? Any papers involving similar questions would be extremely appreciated!

Here's an example: $(x^{2}+2x+1)p(x)+x$ seems to be irreducible for "almost all" p(x) with nonzero constant term. Unsure how I'd show this. Here's a paper that kind of has the flavor of problems that I'm interested in.

Cross-posted from Math.SE.

I've been wondering recently about results for irreducibility that use the "additive structure" of the polynomial ring at hand. For instance, can we say anything about the irreducibility of a sum of two irreducible polynomials that satisfy some conditions? This seems like an additive number theory problem but in a different context. I searched the literature for stuff along these lines but did not find much. Am I missing something obvious (like maybe I can be asking the same questions in the underlying ring) or is this just a question with sparse literature around it? Any papers involving similar questions would be extremely appreciated!

Here's an example: $(x^{2}+2x+1)p(x)+x$ seems to be irreducible for "almost all" p(x) with nonzero constant term. Unsure how I'd show this. Here's a paper that kind of has the flavor of problems that I'm interested in.

Source Link
Dan
Dan
  • 111
  • 4

Results on additive structure of polynomial rings?

I've been wondering recently about results for irreducibility that use the "additive structure" of the polynomial ring at hand. For instance, can we say anything about the irreducibility of a sum of two irreducible polynomials that satisfy some conditions? This seems like an additive number theory problem but in a different context. I searched the literature for stuff along these lines but did not find much. Am I missing something obvious (like maybe I can be asking the same questions in the underlying ring) or is this just a question with sparse literature around it? Any papers involving similar questions would be extremely appreciated!

Here's an example: $(x^{2}+2x+1)p(x)+x$ seems to be irreducible for "almost all" p(x) with nonzero constant term. Unsure how I'd show this. Here's a paper that kind of has the flavor of problems that I'm interested in.