hp pavillon dual boot ubuntu

Enable legacy in the boot options of the bios. Then I installed and configured linux (debian ubuntu and arch linux)

This may not have been the worst thing you could possibly have done, but it was an action that virtually guaranteed you'd run into problems. To learn why, see my Web page on the subject. (Short version: Enabling the CSM, aka "legacy support," turns the boot process into a coin-toss, with "heads" being no different than leaving it disabled and "tails" being a 100% guarantee of problems in dual-booting.)

One fix is to back up and start again -- disable the CSM and re-install all of your Linux distributions. This is overkill, but in many ways it's the easiest solution. Note that if you enabled the CSM because you had problems booting the installation media, chances are they were prepared incorrectly. See the page to which I linked in the previous paragraph for more on that subject.

Another solution is to install an EFI-mode boot loader, which should get your existing installations working. There are many ways to do this, but in all cases you should start by disabling the CSM. Two that are likely to be relatively easy are:

• If it's not already disabled, disable Secure Boot in your firmware. Download the USB flash drive or CD-R version of my rEFInd boot manager and prepare a medium with it. You should be able to boot that, and from it boot any of your Linux distributions. (Some may require some extra coaxing, depending on how you installed them.) In a Linux distribution, install rEFInd to your hard disk. (I provide a Debian package that should work in Debian or Ubuntu and a PPA that should work in Ubuntu. Arch has its own rEFInd package.) This should then get you booting -- but again, you may need to tweak things a bit.
• Run the Boot Repair utility from a boot of an Ubuntu emergency disk, booted in EFI mode. This should install the EFI version of GRUB 2 to your computer, which should pick up your distributions and make them bootable. Unfortunately, GRUB 2 can be a bit iffy with multiple Linux distributions, especially exotic ones -- the OS-detection scripts sometimes aren't optimized or up-to-date for distributions other than the one for which the package was made. Thus, if you use Boot Repair (which is basically an Ubuntu tool), it's quite possible that Arch, and possibly even Debian, will not be correctly detected.

Either method may work partially, but lead you into a new problem. If this happens, post back or start a new question; there are too many potential pitfalls in your quad-boot setup for me to begin to address even a tiny fraction of them pre-emptively.